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Quasilinear asymptotically periodic elliptic equations with critical growth
Nonlinear Analysis 71 (2009) 2890–2905
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Quasilinear asymptotically periodic elliptic equations with critical growth Haendel F. Lins, Elves A.B. Silva ∗ Departamento de Matemática, Universidade de Brasília, 70910-900 Brasília-DF, Brazil
article
abstract
info
Article history: Received 19 November 2008 Accepted 21 January 2009
The existence of a nontrivial solution for a class of asymptotically periodic equations involving the p-Laplacian operator in the euclidean space is established. The solutions are obtained by an application of the Mountain Pass Theorem and the ConcentrationCompactness Principle. The lack of compactness due to the fact that the domain is unbounded is compensated by comparing the minimax level associated with the original problem and the minimax level associated with the periodic limit problem. © 2009 Elsevier Ltd. All rights reserved.
MSC: 35J20 35J60 Keywords: Quasilinear elliptic equations Sobolev critical exponent
1. Introduction In this article, we study the existence of a nontrivial solution for the quasilinear elliptic problem ∗
−∆p u + V (x)up−1 = K (x)up −1 + g (x, u), x ∈ RN , (1.1) u ∈ W 1,p (RN ), u ≥ 0, where ∆p u = div |∇ u|p−2 ∇ u is the p-Laplacian of u, 1 < p < N, p∗ = Np/(N − p) is the critical Sobolev exponent, and V , K : RN → R and g : RN × R → R are continuous functions.
Our main goal is to establish the existence of a nontrivial solution for the Problem (1.1) under an asymptotic periodicity condition at infinity. Considering F the class of functions h ∈ C (RN , R) ∩ L∞ (RN ) such that, for every ε > 0, the set {x ∈ RN : |h(x)| ≥ ε} has finite Lebesgue measure, we suppose that V and K are perturbations of periodic functions at infinity. More specifically, we assume that V and K satisfy
(V ) there exist a constant a0 > 0 and a function V0 ∈ C (RN , R), 1-periodic in xi , 1 ≤ i ≤ N, such that V0 − V ∈ F and V0 (x) ≥ V (x) ≥ a0 > 0, for all x ∈ RN ;
(K ) there exist a function K0 ∈ C (RN , R), 1-periodic in xi , 1 ≤ i ≤ N, and a point x0 ∈ RN , such that K − K0 ∈ F and (i) K (x) ≥ K0 (x) > 0, for all x ∈ RN , N −p (ii) K (x) = |K |∞ + O(|x − x0 | p−1 ), as x → x0 . Rs Considering G(x, s) = 0 g (x, t ) dt, the primitive of g, we also suppose the following hypotheses: (g1 ) g (x, s) = o(|s|p−1 ), as s → 0+ , uniformly in RN ; ∗
Corresponding author. Tel.: +55 61 3273 3356; fax: +55 61 3273 2737. E-mail addresses: [email protected] (H.F. Lins), [email protected] (E.A.B. Silva).
0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.171
H.F. Lins, E.A.B. Silva / Nonlinear Analysis 71 (2009) 2890–2905
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(g2 ) there are constants a1 , a2 > 0 and p < q1 < p∗ such that |g (x, s)| ≤ a1 + a2 |s|q1 −1 ,
for all (x, s) ∈ RN × [0, +∞);
(g3 ) there exist a constant p ≤ q2 < p∗ and functions h1 ∈ L1 (RN ), h2 ∈ F such that 1 p
g (x, s)s − G(x, s) ≥ −h1 (x) − h2 (x)|s|q2 ,
for all (x, s) ∈ RN × [0, +∞).
We observe that the conditions (g1 ) and (g2 ) allow us to employ variational methods to study Problem (1.1) and to verify that the associated functional has a local minimum at the origin. However, under these hypotheses, this functional does not satisfy the Palais–Smale compactness condition, since the domain is unbounded and the Problem (1.1) involves critical growth with respect to the Sobolev Imbedding. The condition (g2 ) imposes a subcritical growth on g. We also observe that the condition (g3 ) is used in the proof of the boundedness of the Palais–Smale sequences of the functional associated with (1.1). The asymptotic periodicity of g at infinity is given by the following condition:
(g4 ) there exist a constant p ≤R q3 < p∗ and functions h3 ∈ F , g0 ∈ C (RN × R, R), 1-periodic in xi , 1 ≤ i ≤ N, such that s (i) G(x, s) ≥ G0 (x, s) = 0 g0 (x, t ) dt, for all (x, s) ∈ RN × [0, +∞), (ii) |g (x, s) − g0 (x, s)| ≤ h3 (x)|s|q3 −1 , for all (x, s) ∈ RN × [0, +∞), ∗ (iii) the function (K0 (x)sp −1 + g0 (x, s))/sp−1 is nondecreasing in the variable s > 0. Finally, we also suppose that g satisfies:
(g5 ) there exists an open bounded set Ω ⊂ RN , containing x0 given by (K )-(ii), such that G(x,s) (i) sp → ∞, as s → ∞, uniformly in Ω , if N > p2 , G(x,s) (ii) sp log s → ∞, as s → ∞, uniformly in Ω , if N = p2 , (iii)
G(x,s)
p∗ −
s
p p−1
→ ∞, as s → ∞, uniformly in Ω , if N < p2 .
We observe that (g5 ) is essential to verify the fact that the mountain pass minimax level is in the interval where the ∗ local convergence in Lp (RN ) for the Palais–Smale sequences of the functional associated with the Problem (1.1) can be established. Now we may state our main result. Theorem A. Suppose that (V ), (K ) and (g1 )–(g5 ) are satisfied. Then Problem (1.1) has a nontrivial solution. The pioneering work on elliptic equations involving critical growth is due to Brézis and Nirenberg [1]. They studied the existence of a positive solution to a semilinear problem with critical growth in a bounded domain of RN , N ≥ 3. Since this famous article, related problems have been one of the main subjects of study in nonlinear analysis (see, for instance, [2–11] and the references therein). The existence of a nontrivial solution for periodic problems involving the Laplace operator and subcritical growth has been established by Rabinowitz [12]. Coti Zelati and Rabinowitz [13] proved that this class of problems has an infinite number of solutions. In the indefinite case (that is, the one in which the potential V can change sign), the existence of a nontrivial solution was established by Troestler and Willem [14] and Kryszewski and Szulkin [15]. The main characteristics of the problem studied in this article are the criticality of the growth and the unboundedness of the domain. Alves, do Ó and Miyagaki, in [16], studied Problem (1.1) with g (x, s) = h(x)sq−1 , where h ∈ C (RN , R) and V , K converging uniformly at infinity to periodic functions. Employing the Concentration-Compactness Principle, appropriate estimates on the mountain pass minimax level and comparison arguments, involving the Nehari manifold, between the minimax levels related with the Problem (1.1) and the periodic corresponding problem, these authors established the existence of nontrivial solutions for (1.1) and the corresponding periodic problem. We also observe that results for the Laplace operator, related to the ones obtained in [16], were established earlier by Alves, Carrião and Miyagaki [17] (see [27– 29] for related results). Finally, we observe that the existence of a nontrivial solution for a periodic problem involving the Laplace operator, critical growth and indefinite potential has been studied by Chabrowski and Yang [18], Chabrowski and Szulkin [19] and do Ó and Ruf [20]. In order to prove Theorem A, we adapt the arguments employed in [17,16]: we suppose, by contradiction, that the only solution for the Problem (1.1) is the trivial one. We use a version of the Mountain Pass Theorem without compactness condition [21] to get a Palais–Smale sequence associated with the minimax level. By appropriate estimates for this level, ∗ we establish, via the Concentration-Compactness Principle, the local convergence in Lp (RN ) of this Palais–Smale sequence. To obtain such estimates, adapting ideas contained in Brézis–Nirenberg [1] and Talenti [22], we use functions which are ∗ extremals for the Sobolev embedding D1,p (RN ) ⊂ Lp (RN ). Next, we use the obtained Palais–Smale sequence and a technical result due to Lions (cf. Coti Zelati and Rabinowitz [13]) to get a nontrivial critical point of the functional associated with the periodic problem. Furthermore, we are able to prove that the value of the functional associated with the Problem (1.1) at this point is less than or equal to the mountain pass minimax level and that this level is attained. We then employ a local
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version of the Mountain Pass Theorem (Theorem 2.3) to obtain a nontrivial critical point of the functional associated with the Problem (1.1). This contradicts our initial assumption that the only possible solution for Problem (1.1) is the trivial one. We observe that in our comparison argument we do not employ the Nehari manifold and, under the hypotheses of Theorem A, we do not know if the periodic problem associated with the Problem (1.1) has a nontrivial solution. However, in the particular case: V = V0 , g = g0 , K = K0 , Theorem A clearly gives us a nontrivial solution for the periodic problem. Indeed, in order to show the existence of a nontrivial solution for the periodic problem, condition (g4 ) is not necessary, that is, considering the problem
∗ −1
−∆p u + V0 (x)up−1 = K0 (x)up u ∈ W 1,p (RN ), u ≥ 0,
+ g0 (x, u),
x ∈ RN ,
(1.2)
under the hypotheses
(V0 ) the function V0 ∈ C (RN , R) is 1-periodic in xi , 1 ≤ i ≤ N, and there exists a constant a0 > 0 such that V0 (x) ≥ a0 > 0,
for all x ∈ RN ;
(K0 ) the function K0 ∈ C (R , R) is 1-periodic in xi , 1 ≤ i ≤ N, and there is a point x0 ∈ RN such that (i) K0 (x) > 0, for allx ∈ RN , N −p (ii) K0 (x) = kK0 k∞ + O(|x − x0 | p−1 ), as x → x0 ; N
and the function g0 satisfying (g1 )–(g3 ) and (g5 ), we may state: Theorem B. Suppose that (V0 ), (K0 ), (g1 )–(g3 ) and g5 are satisfied. Then Problem (1.2) possesses a nontrivial solution. 2. Preliminary results In this section we present two versions of the Mountain Pass Theorem due to Ambrosetti and Rabinowitz [23], which are essential tools in this article. We also introduce the variational framework associated with the Problem (1.1) and show the boundedness of the Palais–Smale sequences of the associated functional. 2.1. Versions of the Mountain Pass Theorem Let E be a real Banach space and I : E → R a C 1 functional. Let K be the set of critical points of I. Given b ∈ R, we define I b = {u ∈ E : I (u) ≤ b} and Kb = {u ∈ E : u ∈ K , I (u) = b}. The open ball in E with center at point x and radius r > 0 is denoted by Br (x). The symbol ∂ Br (x) denotes the boundary of this ball. Unfortunately, the functional associated with the Problem (1.1) does not satisfy the Palais–Smale condition. To overcome this difficulty, we use versions of the Mountain Pass Theorem. In the following, we state the first version of this Theorem (see also [23]). Theorem 2.1. Let E be a real Banach space. Suppose I ∈ C 1 (E , R) is such that I (0) = 0 and
(I1 ) there exist ρ, α > 0 such that I |∂ Bρ (0) ≥ α > 0, (I2 ) there exists e ∈ E \Bρ (0) such that I (e) ≤ 0. Then I possesses a (PS )c sequence for c ≥ α > 0 given by c = inf max I (γ (t )),
(2.1)
Γ = {γ ∈ C ([0, 1], E ) : γ (0) = 0, γ (1) ∈ I 0 ∩ (E \Bρ (0))}.
(2.2)
γ ∈Γ t ∈[0,1]
where
Proof. Supposing by contradiction that the theorem is false, it follows that the condition (PS )c is vacuously satisfied. By the Mountain Pass Theorem [24], there exists a critical point u ∈ E on level c, that is, I (u) = c and I 0 (u) = 0. Then the sequence (um ) given by um = u for all m ∈ N is a (PS )c sequence for I. This is a contradiction, and the proof of Theorem 2.1 is complete. We will also need to establish a local version of Theorem 2.1. In order to show this fact, we first state a local deformation result, given by the lemma below. Lemma 2.2. Let E be a real Banach space. Assume that I ∈ C 1 (E , R) and, for some b ∈ R, there exists a compact set D ⊂ I b such that D 6= ∅ and D ∩ Kb = ∅. Then, given ε¯ > 0, there exist 0 < ε < ε¯ and η ∈ C ([0, 1] × E , E ) such that
(η1 ) η(t , u) = u, for all t ∈ [0, 1], I (u) 6∈ [b − ε¯ , b + ε¯ ], (η2 ) I (η(t , u)) ≤ I (u) for all u ∈ E and t ∈ [0, 1], (η3 ) η(1, D) ⊂ I b−ε .
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Theorem 2.3. Let E be a real Banach space. Suppose that I ∈ C 1 (E , R) satisfies I (0) = 0, (I1 ) and (I2 ). If there exists γ0 ∈ Γ , Γ defined by (2.2), such that c = max I (γ0 (t )) > 0,
(2.3)
t ∈[0,1]
then I possesses a nontrivial critical point u ∈ Kc ∩ γ0 ([0, 1]). Proof. Arguing by contradiction, suppose that γ0 ([0, 1]) ∩ Kc = ∅. Thus, let D = γ0 ([0, 1]) and 0 < ε¯ < c and apply Lemma 2.2 to find 0 < ε < ε¯ and η ∈ C ([0, 1] × E , E ) satisfying conditions (η1 )–(η3 ). Taking γ (t ) = η(1, γ0 (t )) for all t ∈ [0, 1], it follows from I (0) = 0, (I2 ), (η1 ) and our choice of ε¯ that γ ∈ Γ . Furthermore, by (2.3) and (η3 ), max I (γ (t )) ≤ c − ε.
t ∈[0,1]
However, this inequality contradicts the definition of c. The proof of the theorem is concluded.
2.2. The variational framework Consider the Sobolev space E = W 1,p (RN ) endowed with one of the following norms p
Z
|∇ u|p + V (x)|u|p ,
k uk = RN
kukp0 =
Z
|∇ u|p + V0 (x)|u|p .
RN
Note that, in view of (V ), the above norms are both equivalent to the standard norm on E. The functional I : E → R associated with the Problem (1.1) is given by I (u) =
1 p
k uk p −
Z
1 p∗
∗
RN
K (x)|u+ |p −
Z RN
G(x, u),
(2.4)
where, in order to obtain a nonnegative solution, we set g (x, s) = 0, for x ∈ RN , s < 0. Under the hypotheses (V ), (K ), (g1 ) and (g2 ) we have that I ∈ C 1 (E , R). Furthermore, the critical points of this functional are the weak solutions for the Problem (1.1). In a similar fashion, the functional I0 ∈ C 1 (E , R), defined by I0 (u) =
1 p
p 0
k uk −
1
Z
p∗
+ p∗
RN
K0 (x)|u |
Z − RN
G0 (x, u),
(2.5)
is associated with the periodic problem. We also suppose that g0 (x, s) = 0, for x ∈ RN , s < 0. Observe that, given δ > 0, in view of (g1 ) and (g2 ), we may find a constant Cδ > 0 such that
|g (x, s)| ≤ δ|s|p−1 + Cδ |s|q1 −1 , for all (x, s) ∈ RN × R, δ Cδ |G(x, s)| ≤ |s|p + |s|q1 , for all (x, s) ∈ RN × R. p
q1
(2.6) (2.7)
These estimates and a standard argument (see [24]) show that the functional I satisfies the geometric conditions (I1 ) and (I2 ). Since the proof of this fact is standard, we just state it: Lemma 2.4. Suppose that (V ), (K ), (g1 ) and (g2 ) are satisfied. Then the functional I defined by (2.4) satisfies I (0) = 0, (I1 ) and (I2 ). As a consequence of Theorem 2.1 and Lemma 2.4, we have Corollary 2.5. Suppose that (V ), (K ), (g1 ) and (g2 ) are satisfied. Then the functional I possesses a (PS )c sequence, with c given by (2.1). As a final result of this section, we verify the boundedness of the (PS ) sequences associated to the functional I. First, however, we establish a simple result that will be employed several times in our work. From now on, |E | will denote the Lebesgue measure of a measurable set E ⊂ RN . In the following lemma, given h ∈ F , we set Dε = {x ∈ RN : |h(x)| ≥ ε} and Dε (R) = {x ∈ RN : |h(x)| ≥ ε, |x| ≥ R}. Lemma 2.6. Suppose h ∈ F . Then |Dε (R)| → 0 as R → ∞, Proof. Since h ∈ F , |Dε | < ∞ for all ε > 0. In order to prove the lemma, it suffices to verify the following claim: lim |Dε ∩ (RN \BRm )| = 0
m→∞
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for each sequence (Rm ) ⊂ R such that Rm → ∞. Consider the real function f : RN → R given by f (x) = χDε (x), that is, f (x) =
for x ∈ Dε for x 6∈ Dε .
1 0
Then f ∈ L1 (RN ) and |f |1 = RN |f | = |Dε |. Moreover, defining the sequence of functions fm : RN → R by fm (x) = χDε ∩(RN \BRm ) (x), it follows that |fm | ≤ |f |. Since fm → 0 almost everywhere in RN as m → ∞, our claim follows from Lebesgue’s Dominated Convergence Theorem.
R
Lemma 2.7. Suppose that (V ), (K ), (g1 ), (g2 ) and (g3 ) are satisfied. Then every (PS ) sequence is bounded. Proof. Let (um ) ⊂ E be a (PS )b sequence, with b ∈ R. Arguing by contradiction, we suppose that (um ) is not bounded. Without loss of generality, kum k → ∞ as m → ∞. First, we claim that there exists λ2 > 0 such that, for m sufficiently large, we have ∗
kum kp ≤ λ2 |um |pp∗ .
(2.8)
Indeed, since I (um ) → 0 strongly in E , we may write 0
∗
k um k p =
Z RN
Z
g (x, um )um +
∗
RN
K (x)|um |p + o(kum k).
(2.9)
Given δ > 0, we invoke (2.6), (2.9) and the Sobolev Embedding Theorem to find C1 > 0 and C2 = C2 (δ) > 0 such that ∗
kum kp ≤ δ C1 kum kp + C2 |um |qq11 + |K |∞ |um |pp∗ + o(kum k). Thus, since p > 1, taking δ > 0 sufficiently small and m sufficiently large, we obtain ∗
kum kp ≤ 2C2 |um |qq11 + 2|K |∞ |um |pp∗ .
(2.10)
Now, setting 0 < β < 1 such that q1 = β p + (1 − β)p , we apply Young’s inequality to find ∗
∗
|um |qq11 ≤ βε1/β |um |pp + (1 − β)ε −1/(1−β) |um |pp∗ ,
(2.11)
for all ε > 0. Choosing ε > 0 sufficiently small, we apply (2.10), (2.11) and the Sobolev Embedding Theorem to find λ2 > 0 such that (2.8) holds. The claim is proved. Now, invoking (g3 ), we have I ( um ) −
1 0 1 I (um ).um = p N
≥
1
Z
∗
N
ZR
N
RN
K (x)|um |p +
Z RN
1 p
g (x, um )um − G(x, um )
p∗
K (x)|um | dx − |h1 |1 −
Z RN
h2 (x)|um |q2 .
Thus, again using the fact that (um ) is a (PS )b sequence, we find C3 > 0 such that 1 N
Z
∗
RN
K (x)|um |p ≤ C3 +
Z RN
h2 (x)|um |q2 + o(kum k).
(2.12)
Given ε > 0, we set Dε (R) = {x ∈ RN : |x| ≥ R, |h2 (x)| ≥ ε}, for all R > 0. Then, since h2 ∈ F , applying Lemma 2.6, we find R = Rε > 0 such that |Dε (R)| < ε . By Hölder’s inequality,
Z Dε (R)
∗ ∗ q h2 |um |q2 ≤ |h2 |∞ ε (p −q2 )/p |um |p2∗
(2.13)
and
Z RN \D
ε (R)
h2 |um |q2 dx ≤ ε|um |qq22 + |h2 |∞
wN RN
(p∗ −q2 )/p∗
N
q
|um |p2∗ .
Furthermore, considering 0 < γ ≤ 1 such that q2 = γ p + (1 − γ )p∗ , we may invoke Hölder’s inequality, the Sobolev p∗
q
Embedding Theorem and (2.8) to find C4 > 0 such that |um |q22 ≤ C4 |um |p∗ . Consequently,
Z
q2
RN \D
ε (R)
h2 |um |
p∗ C4 um p∗
≤ε |
| + |h2 |∞
wN RN N
(p∗ −q2 )/p∗
q
|um |p2∗ .
(2.14)
Choosing ε > 0 sufficiently small, we use (2.8), (2.12)–(2.14), condition (K ) and the fact that q2 < p∗ to conclude that (um ) ∗ is bounded in Lp (RN ). Furthermore, using (2.8) one more time, it follows immediately that (um ) is also bounded in E. The proof of Lemma 2.7 is complete.
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3. Concentration compactness In this section we state the Concentration-Compactness Principle due to Lions [25], which is fundamental to overcome the lack of compactness caused by the presence of a critical term in the Problem (1.1), and establish some related results. We also present results concerning the behaviour of the Palais–Smale sequences: we pointed out the strong convergence in p−N p
p∗
N
Lloc (RN ) of (PS )b sequences when b < N1 |K |∞ S p and a proposition which will be essential to guarantee that the solution that we provide in our proof of Theorem A is nontrivial. 3.1. The Concentration-Compactness Principle We first recall that the best constant for the Sobolev embedding D1,p (RN ) ⊂ Lp (RN ) is given by ∗
R S=
inf
u∈D1,p (RN )
RN
u6=0
|∇ u|p p/p∗ . |u|p∗
RN
R
(3.1)
Now we state the Concentration-Compactness Principle due to Lions [25]. Lemma 3.1. Suppose 1 ≤ p < N and let (um ) ⊂ D1,p (RN ) be a sequence such that um * u weakly in D1,p (RN ) and ∗ |∇ um |p dx * µ, |um |p dx * ν weakly in the sense of measures, where µ and ν are nonnegative bounded measures in RN . Then there exist a set at most countable J and a family {xj : j ∈ J } of points in RN such that ∗
(a) ν = |u|p dx + j∈J νj δxj , P (b) µ ≥ |∇ u|p dx + j∈J µj δxj ,
P
p/p∗
where {νj : j ∈ J } and {µj : j ∈ J } are two families of positive numbers such that S νj
P
j∈J
p/p∗ j
ν
≤ µj for all j ∈ J. In particular,
< ∞. ∗
p P Given K by condition (K ) and the measure ν established by Lemma 3.1, we denote by K ν the measure K |u| dx + K ( x )ν δ , then we state some auxiliary results. j j x j∈J j
Lemma 3.2. Suppose that (V ), (K ), (g1 ) and (g2 ) are satisfied. If um * u weakly in E, where xj , νj , j ∈ J are given by Lemma 3.1, we have: V |um |p → V |u|p in L1loc (RN ), g (x, um )um → g (x, u)u in
L1loc
(3.2)
(R ), N
(3.3)
p∗
K |um | dx * K ν weakly in the sense of measures.
(3.4)
In the following, we refer the reader to the references [2] and [7]. We fix ϕ ∈ C0∞ (RN , [0, 1]), ϕ ≡ 1 in B1/2 (0), ϕ ≡ 0 in RN \B1 (0). Given ε > 0, j ∈ J and xj ∈ RN as in Lemma 3.1, consider ϕε = ϕεj : RN → R defined by
ϕε (x) = ϕ
x − xj
ε
,
for every x ∈ RN .
(3.5)
Lemma 3.3. Suppose that (K ) is satisfied. Let µ and ν be as in Lemma 3.1 and 1 ≤ q < ∞. Then we have
Z RN
ϕεq dν → νj ,
Z RN
ϕεq dµ → µj ,
Z RN
K ϕεq dν → K (xj )νj ,
as ε → 0.
Lemma 3.4. Suppose that (V ), (K ), (g1 ) and (g2 ) are satisfied. Let (um ) ⊂ E be a bounded sequence satisfying um * 0 weakly in E, I 0 (um ) → 0 strongly in E ∗ , as m → ∞. Then, letting xj , νj , j ∈ J be as in Lemma 3.1, we have J = ∅ or νj ≥ all j ∈ J .
S K (xj )
N /p
3.2. Behaviour of the Palais–Smale sequences We start with the following auxiliary result: Lemma 3.5. Suppose that (g1 )–(g3 ) are satisfied. Let (um ) ⊂ E be a bounded sequence such that um * 0 weakly in E. Then
Z lim inf m→∞
RN
1 p
g (x, um )um − G(x, um ) ≥ 0.
for
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Proof. Arguing by contradiction, we suppose that there exist a subsequence, still denoted by (um ), a > 0 and m1 ∈ N such that
Z
1 p
RN
g (x, um )um − G(x, um ) ≤ −a < 0,
(3.6)
for all m ≥ m1 . Now, by the condition (g3 ),
Z
1 p
RN \BR (0)
Z
g (x, um )um − G(x, um )
Z
dx ≥ − RN \BR (0)
h1 −
RN \BR (0)
h2 |um |q2 ,
(3.7)
for all R > 0, with q2 ∈ (p, p∗ ). Given η > 0, since h1 ∈ L1 (RN ), there exists R0 > 0 such that, for R ≥ R0 ,
Z
|h1 | <
RN \BR (0)
η 4
.
(3.8)
Taking δ > 0 to be chosen later, define Dδ (R) = {x ∈ RN : |h2 (x)| ≥ δ, |x| ≥ R}. Since h2 ∈ F , we may choose R1 > 0 such that |Dδ (R)| < δ for all R ≥ R1 . But,
Z RN \BR (0)
h2 |um |q2 =
Z Dδ (R)
h2 |um |q2 +
Z δ (R)∪BR (0))
RN \(D
h2 |um |q2 ,
since Dδ (R) ⊂ R \BR (0). In view of Hölder’s inequality and our choice of R1 , we have, for all R ≥ R1 , N
Z
q2
Dδ (R)
|h2 ||um |
Z ≤ |h2 |∞
p∗
Dδ (R)
q∗2 p
|um |
|Dδ (R)|(p
∗ −q )/p∗ 2
≤ c1 δ (p
∗ −q )/p∗ 2
,
and
Z RN \(Dδ (R)∪BR (0))
|h2 ||um |q2 ≤ δ
Z RN
|um |q2 ≤ c2 δ,
where c1 and c2 are positive constants, obtained via the Sobolev Embedding Theorem and the boundedness of (um ) ⊂ E. Thus
Z RN \BR (0)
|h2 ||um |q2 ≤ c1 δ
p∗ −q2 p∗
+ c2 δ.
Therefore, we may choose δ > 0 such that, for R ≥ R1 > 0,
Z RN \BR (0)
|h2 ||um |q2 <
η 4
.
This inequality, (3.7) and (3.8), give us
Z RN \BR (0)
1 p
η g (x, um )um − G(x, um ) ≥ − ,
(3.9)
2
for every R ≥ max{R0 , R1 }. Now, for such R, from (g1 ) and (g2 ), there exists c3 > 0 such that
Z
BR (0)
Z g (x, um )um − G(x, um ) ≤ p
1
BR (0)
|um |p + c3 |um |q1 ,
where q1 ∈ (p, p ) is given by (g2 ). Since um * 0 weakly in E, taking a subsequence if necessary, we may suppose that q1 um → 0 in Lloc (RN ). Thus there exists m0 = m0 (η) ∈ N such that, for m ≥ m0 , ∗
Z BR (0)
1
η
g (x, um )um − G(x, um ) ≥ − . p 2
(3.10)
By our choice of R, (3.9) and (3.10), we have
Z RN
1 p
g (x, um )um − G(x, um ) ≥ −η,
for all m ≥ m0 . This contradicts (3.6), since η < a, and concludes the proof of the lemma.
H.F. Lins, E.A.B. Silva / Nonlinear Analysis 71 (2009) 2890–2905
2897
Proposition 3.6. Suppose that (V ), (K ) and (g1 )–(g3 ) are satisfied. Let (um ) ⊂ E be a (PS )b sequence with 0 < b < 1 N
(p−N )/p N /p
|K |∞
p∗
and um * 0 weakly in E. Then, up to a subsequence, um → 0 strongly in Lloc (RN ) as m → ∞.
S
Proof. In view of Lemma 3.1, taking a subsequence if necessary, we have ∗
|um |p dx * ν =
X
νj δxj ,
j∈J
|∇ um |p dx * µ ≥
X
µj δ x j ,
j∈J
weakly in the sense of measures, where J is a set at most countable and νj , µj > 0 for every j ∈ J . We claim that J = ∅. Arguing by contradiction, suppose that there exists j0 ∈ J. Since I (um ) → b < I 0 (um ) → 0 and (um ) ⊂ E is bounded, we find m1 ∈ N such that p−N
1
p−N
1 N
N
|K |∞p S p ,
1
N
|K |∞p S p > I (um ) − I 0 (um ).um N p Z Z 1 1 ∗ K |um |p + = g (x, um )um − G(x, um ) , N
RN
p
RN
for all m ≥ m1 . In view of the Lemma 3.5, taking a greater m1 if necessary, we may assume that 1 N
p−N
N
1
1
p
N
|K |∞p S p > I (um ) − I 0 (um ).um ≥
Z
∗
RN
K |um |p ,
(3.11)
for all m ≥ m1 . Now, taking ϕ ∈ C0∞ (RN , [0, 1]) such that ϕ(xj0 ) = 1 and invoking (3.11), it follows that 1 N
p−N
1
N
|K |∞p S p >
N
Z
∗
RN
ϕ K |um |p dx,
for all m ≥ m1 . Letting m → ∞, in view of (3.11) and Lemmas 3.2 and 3.4 we get 1 N
p−N
1 X
N
|K |∞p S p >
N j∈J
1
K (xj )νj ϕ(xj ) ≥
N
K (xj0 )νj0 ≥
1 N
K (xj0 )
p−N p
N
Sp.
This contradicts K (xj0 ) ≤ |K |∞ . The claim is proved. ∗
Then, we may conclude that |um |p dx * 0 weakly in the sense of measures in RN . Indeed, given a compact set L ⊂ RN , we choose ϕ ∈ C0∞ (RN , [0, 1]) such that ϕ ≡ 1 em L to get
Z
∗
|um |p ≤
0≤
Z
∗
RN
L
|um |p ϕ → 0,
as m → ∞.
This concludes the proof of the proposition.
p−N p
N
Proposition 3.7. Suppose that (V ), (K ), (g1 ) and (g2 ) are satisfied. Let (um ) ⊂ E be a (PS )b sequence with 0 < b < N1 |K |∞ S p , and um * 0 weakly in E as m → ∞. Then there exist a sequence (ym ) ⊂ RN and r , η > 0 such that |ym | → ∞ and
Z lim sup Br (ym )
m→∞
|um |p ≥ η > 0.
Proof. Supposing that the result does not hold, we have (see [13] or [26]):
Z RN
|um |σ → 0,
for every σ ∈ (p, p∗ ).
(3.12)
Since (um ) ⊂ E is bounded and I 0 (um ) → 0 in E ∗ , we get
k um k p −
Z RN
g (x, um )um −
Z
∗
RN
K |um |p = om (1).
(3.13)
We claim that
Z lim
m→∞
RN
g (x, um )um = 0.
(3.14)
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H.F. Lins, E.A.B. Silva / Nonlinear Analysis 71 (2009) 2890–2905
Indeed, from (2.6), (3.12) and the fact that (um ) ⊂ E is bounded, we find C2 > 0 such that
Z
|g (x, um )um | ≤ δ C2 .
lim sup RN
m→∞
Since δ > 0 can be chosen sufficiently small, the above inequality establishes (3.14), proving the claim. Similarly, we conclude that
Z lim
m→∞
RN
G(x, um ) = 0.
(3.15)
Since b > 0, we may not have um → 0 strongly in E. Thus, taking a subsequence if necessary, kum kp → l > 0 as m → ∞. From (3.13) and (3.14), we get
Z
∗
RN
K |um |p → l. p−N
Since I (um ) → b < b=
1
(3.16)
1 N
N
|K |∞p S p , in view of (3.15) and (3.16), we obtain
l.
N
(3.17)
But, from the definition of S,
Z
p
k um k ≥
Z
S
p
|∇ um | ≥
K |um |
p
∗
RN
p∗
RN
p |K |∞
p p∗
.
Letting m → ∞ and using (3.16) and (3.17), we get S
bN ≥
(bN )p/p . ∗
p/p∗
|K |∞ Consequently, b≥
1 N
|K |(∞p−N )/p S N /p ,
contradicting the fact that b <
1 N
(p−N )/p N /p
|K |∞
S
. The proposition is proved.
4. Estimates In this section we shall verify that the minimax level associated with the Mountain Pass Theorem (Theorem 2.1) is in ∗ the interval where there is local convergence in Lp (RN ) of the Palais–Smale sequences for the functional associated with (1.1) and the Proposition 3.7 can be applied. To show this result, we use test functions as the ones employed by Brézis and Nirenberg [1], and we verify some auxiliary results about these functions. Then we show the main result of the section, providing the estimate for the minimax level c given by (2.1). 4.1. Test functions Without loss of generality, we assume that x0 , given by the condition (K ), is the origin of RN and that B2 = B2 (0) ⊂ Ω , with Ω given by the condition (g5 ). Given ε > 0, we consider the function wε : RN → R defined by
ε (N −p)/p , (ε + |x|p/(p−1) )(N −p)/p 2
wε (x) = C (N , p) where C (N , p) =
" N
N −p p−1
p−1 #(N −p)/p2
.
We observe that {wε }ε>0 is a family of functions on which the infimum that defines the best constant, S, for the Sobolev ∗ embedding D1,p (RN ) ⊂ Lp (RN ) is attained. We also consider φ ∈ C0∞ (RN , [0, 1]), φ ≡ 1 in B1 , φ ≡ 0 in RN \B2 and define uε uε = φwε , vε = R . ∗ ( RN Kupε )1/p∗
H.F. Lins, E.A.B. Silva / Nonlinear Analysis 71 (2009) 2890–2905
2899
The following lemmas were inspired by [2,8]. We just state them, since their proofs are standard. Lemma 4.1. Suppose that (K ) is satisfied. Then, there exist positive constants k1 , k2 and ε0 such that
Z RN \B1
k1 <
|∇ uε |p = O(ε (N −p)/p ),
Z
∗
RN
Z
Kupε < k2 ,
as ε → 0+ ,
for all 0 < ε < ε0 ,
|x|(N −p)/(p−1) wεp = O(ε(N −p)/p ), ∗
|x|≤1
Z RN
(4.1)
(4.2)
as ε → 0+ ,
|∇vε |p ≤ |K |(∞p−N )/N S + O(ε(N −p)/p ),
(4.3)
as ε → 0+ .
(4.4)
Lemma 4.2. Suppose that (K ) is satisfied. Then, as ε → 0, we have N −p
p
(i) |vε |p = O(ε p ), if N < p2 ; p (ii) |vε |p = O ε p−1 | log ε| , if N = p2 ; p
(iii) |vε |p = O ε p−1 , if N > p2 .
4.2. Estimates for the minimax level Before stating the main result of this section, we present a useful lemma. Lemma 4.3. Suppose that (V ), (K ), (g1 ) and (g2 ) are satisfied. Consider tε > 0 such that I (tε vε ) = maxt ≥0 I (t vε ). Then, there exist ε0 > 0 and positive constants T1 and T2 such that T1 ≤ tε ≤ T2 for every 0 < ε < ε0 . Proof. First, we observe that, in view of Lemma 2.4, I satisfies I (0) = 0 and (I1 ). Consequently, since I ∈ C 1 (E , R) and kvε k is bounded, we conclude that there exists T1 > 0 such that tε ≥ T1 > 0 for every R ε > ∗0 sufficiently small. On the other hand, the estimate tε ≤ T2 < ∞ is obtained using the definition of I, the fact that RN K vεp = 1, (2.7) and the boundedness of kvε k. This proves the lemma. Next we state a result which provides an appropriate estimate on the minimax level. Proposition 4.4. Suppose that (V ), (K ), (g1 ), (g2 ) and (g5 ) are satisfied. Then there exists v ∈ E \{0} such that max I (t v) < t ≥0
1 N
p−N
N
|K |∞p S p .
Proof. Consider tε as defined by Lemma 4.3. Letting Xε to be the integral
R
RN
|∇vε |p , we get
∗
tp tp I (tε vε ) ≤ ε (Xε + |V |∞ |vε |pp ) − G(x, tε vε ) − ε∗ N p p Z R 1 N /p p ≤ Xε + c1 |vε |p − G(x, tε vε ), N RN
Z
for some constant c1 > 0. Indeed, considering the function h = hε : [0, ∞) → R given by h(t ) = 1/(p∗ −p)
that t0 = Xε
I (tε vε ) ≤
1 N
is a maximum point of h and h(t0 ) = p−N
N
(|K |∞N S + O(ε (N −p)/p )) p + c1 |vε |pp −
1 N /p X . N ε
Z RN
1 X tp p ε
−
1 p∗ t , p∗
we have
It follows from (4.4) that
G(x, tε vε ).
Applying the inequality
(b + c )ζ ≤ bζ + ζ (b + c )ζ −1 c ,
b, c ≥ 0, ζ ≥ 1,
we get I (tε vε ) ≤
1 N
|K |(∞p−N )/p S N /p + c1 |vε |pp −
Z RN
G(x, tε vε ) + O(ε
N −p p
).
(4.5)
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Now consider
N −p ε p , γ (ε) = ε p−1 | log ε|, p−1 ε ,
if N < p2 , if N = p2 , if N > p2 .
(4.6)
In view of Lemma 4.2, (4.5) and (4.6), we find a constant M > 0 such that I (tε vε ) ≤
1 N
|K |(∞p−N )/p S N /p + γ (ε) M −
Z
1
γ (ε)
RN
G(x, tε vε ) .
(4.7)
In order to prove Proposition 4.4, we just need to verify that lim
ε→0+
Z
1
γ (ε)
RN
G(x, tε vε ) = +∞.
(4.8)
By (g1 ) and (g5 ), it is possible to choose c > 0 such that G(x, s) +
1 p
csp ≥ 0,
for every x ∈ Ω , s ≥ 0.
(4.9)
Given A0 > 0, we invoke (g5 ) again to obtain R = R(A0 ) > 0 such that, for x ∈ Ω , s ≥ R,
G(x, s) ≥
1 A0 sµ , p 1 p 1 p
if N < p2 ,
A0 sp log s,
if N = p2 ,
A0 sp ,
if N > p2 ,
(4.10)
where µ = p∗ − p/(p − 1). Now consider the function ηε : [0, ∞) → R defined by
ε (N −p)/p . (ε + r p/(p−1) )(N −p)/p 2
ηε (r ) =
Since φ ≡ 1 in B1 , in view of (4.2), we find a constant c1 > 0 such that vε (x) ≥ c1 ηε (|x|) for |x| < 1. Furthermore, since ηε is decreasing, there exists a positive constant α˜ such that, for |x| < ε tε vε (x) ≥ T1 c1 ηε (|x|) ≥ T1 c1 ηε (ε
p−1 p
) ≥ αε ˜
N −p (1−p) p2
p−1 p
,
.
Here T1 is given by Lemma 4.3. Then we may choose ε1 > 0 such that tε vε (x) ≥ αε ˜ for |x| < ε
p−1 p
N −p (1−p) p2
≥ R = R(A0 ),
(4.11)
, 0 < ε < ε1 . It follows from (4.10) and (4.11) that
1 A0 tεµ vεµ , p 1 A0 tεp vεp log(tε vε ), G(x, tε vε (x)) ≥ p 1 A0 tεp vεp , p
if N < p2 , if N = p2 ,
(4.12)
if N > p2 ,
p−1
for |x| < ε p , 0 < ε < ε1 . Now, in order to verify (4.8), we should consider the three possibilities given by (g5 ). However, since they are similar, we will just consider the case N < p2 . Since B2 (0) ⊂ Ω , we invoke (4.9) and (4.12), to get
Z RN
G(x, tε vε ) =
Z
G(x, tε vε ) + B (p−1)/p ε
≥
1 p
A0 tεµ
Z
Z Ω \B (p−1)/p ε
c
B (p−1)/p ε
vεµ − tεp p
G(x, tε vε )
Z Ω \B (p−1)/p ε
vεp
H.F. Lins, E.A.B. Silva / Nonlinear Analysis 71 (2009) 2890–2905
≥
1 p
Z
µ
2901
1
A0 T1
B (p−1)/p ε
vεµ − cT2p |vε |pp , p
for 0 < ε < ε1 , where T1 and T2 are given by Lemma 4.3. Consequently, by Lemma 4.2, there exists M1 > 0 such that
Z RN
G(x, tε vε ) ≥
1 p
Z
µ
A0 T1
B (p−1)/p ε
vεµ − M1 ε
Now, considering the change of variable s = ε
Z B (p−1)/p ε
vεµ (x) dx
µ
Z
≥ c1
B (p−1)/p ε
1−p p
N −p p
µ
(4.13)
= c1 ωN
ε
Z
p−1 p
ε
N −p µ N −1 p2
r
N −p p µ dr p p − 1 ε+r
0
s
N −1
ds ε
p
1 + s p−1
= c2 ε
.
t,
ηεµ (|x|) dx
Z 1 µ = c1 ωN 0
N −p p
N −p p µ
(N −p)(p−1) p2
(p∗ −µ)
,
since µ = p − p/(p − 1). Thus, by (4.13), ∗
1
ε
N −p p
Z RN
G(x, tε vε ) ≥
c2 µ T p 1
A0 − M1 .
Since A0 > 0 can be chosen arbitrarily large, the above relation establishes (4.8). The Proposition 4.4 is proved.
5. Proofs of Theorems A and B In this section we prove Theorems A and B by verifying that the functionals I and I0 , defined by (2.4) and (2.5), respectively, have nontrivial critical points. 5.1. Proof of Theorem A Before proving Theorem A, we introduce two technical results. Lemma 5.1. Suppose that (V ), (K ) and (g4 ) are satisfied. Let (um ) ⊂ E be a bounded sequence and wm (y) = w(y − ym ), where w ∈ W 1,p (RN ) and (ym ) ⊂ RN . If |ym | → ∞, then we have ∗
(K0 − K )|um |p −1 wm → 0, (V0 − V )|um |p−1 wm → 0, (g0 (x, um ) − g (x, um ))wm → 0, strongly in L1 (RN ), as m → ∞. Proof. Considering that the other two limits follow by a similar argument, we will establish the first limit in Lemma 5.1. ∗ Given δ > 0, since w ∈ Lp (RN ), we find 0 < ε < δ such that, for every measurable A ⊂ RN satisfying |A| < ε ,
Z
∗
|w|p < δ.
(5.1)
A
We fix ε > 0 and set Dε (R) = {x ∈ RN : |K (x) − K0 (x)| ≥ ε, |x| ≥ R}. By the condition (K ) and the Lemma 2.6, we may find R > 0 such that |Dε (R)| < ε . Then, by Hölder’s inequality, condition (K ), (5.1) and the fact that (um ) ⊂ E is bounded, we obtain a constant C1 > 0 such that
Z
∗ −1
RN \BR (0)
p |K (x) − K0 (x)||u+ m|
∗
1 |wm | ≤ 2kK k∞ |um |pp∗− |wm |Lp∗ (Dε (R)) + ε|um |pp∗ −1 |w|p∗ ∗ 1
≤ C1 (δ p∗ + δ).
(5.2)
On the other hand, by Hölder’s inequality, the condition (K ) and the fact that (um ) ⊂ E is bounded, we find C2 > 0 such that
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H.F. Lins, E.A.B. Silva / Nonlinear Analysis 71 (2009) 2890–2905
Z
∗ −1
BR (0)
p |K (x) − K0 (x)||u+ m|
∗
|wm | ≤ 2kK k∞ |um |pp∗ −1 |w(. − ym )|Lp∗ (BR (0)) Z ≤ C2
∗
BR (−ym )
|w(y)|p
1/p∗
.
∗
Thus, since w ∈ Lp (RN ) and |ym | → ∞, there exists m0 ∈ N such that
Z
∗ −1
BR (0)
p |K (x) − K0 (x)||u+ m|
|wm | ≤ C2 δ,
for all m ≥ m0 .
(5.3)
p∗ −1 The inequalities (5.2), (5.3) and the fact that δ > 0 can be chosen arbitrarily small imply that (K (x)− K0 (x))|u+ wm → 0 m| 1 N strongly in L (R ) as m → ∞. The proof of Lemma 5.1 is complete.
In order to prove Theorem A we also need of the following result. Lemma 5.2. Suppose p ≤ q < p∗ and h ∈ F . Let (um ) ⊂ E be a sequence such that um * u weakly in E as m → ∞. Then h(x)|um |q → h(x)|u|q strongly in L1 (RN ),
as m → ∞.
Proof. Arguing by contradiction, we suppose that there exist a subsequence, still denoted by (um ), and ε > 0 such that
Z RN
|h(x)| · | |um |q −|u |q | ≥ ε > 0,
for all m ∈ N.
(5.4)
Now we define Dδ (R) = {x ∈ RN : |h(x)| ≥ δ, |x| ≥ R}. Since h ∈ F , there exists R = Rδ > 0 such that |Dδ (R)| < δ . Consequently, applying Hölder’s inequality, we get
Z Dδ (R)
|h(x)| · | |um |q −|u |q | ≤ |h|∞
Z Dδ (R)
(|um |q + |u|q )
"Z ≤ |h|∞
|um |
p∗
q p∗
Z
p∗
+
| u| RN
RN
q∗ # p
|Dδ (R)|
p∗ −q p∗
≤ C1 δ
p∗ −q p∗
,
(5.5)
for some constant C1 > 0. On the other hand, by the definition of Dδ (R),
Z
|h(x)| · | |um | −|u | | ≤ δ q
BCR \Dδ (R)
q
Z RN
(|um |q + |u|q ) ≤ C2 δ,
(5.6)
for some constant C2 > 0. Observe that the constants C1 and C2 are obtained via the Sobolev Embedding Theorem and the boundedness of (um ). Since um * u weakly in E, it follows from the Sobolev Embedding Theorem that, up to a subsequence, q um → u in Lloc (RN ). In particular, for R = Rδ > 0 given above, without loss of generality, we may suppose that
Z
| |um |q −|u |q | → 0,
as m → ∞.
BR
Then, taking into account the fact that h ∈ L∞ (RN ), there exists m0 ∈ N such that
Z
|h(x)| · | |um |q −|u |q | < δ,
for all m ≥ m0 .
(5.7)
BR
Thus, it follows from (5.5)–(5.7) that, for m ≥ m0 = m0 (δ),
Z RN
|h(x)| · | |um |q −|u |q | < C1 δ
p∗ −q p∗
+ (1 + C2 )δ.
Since δ > 0 can be chosen arbitrarily small, the above relation contradicts (5.4). This proves the Lemma 5.2.
Now, we may conclude the proof of Theorem A. First, we invoke Corollary 2.5 to find a sequence (um ) ⊂ E such that I ( um ) → c ≥ α > 0
and kI 0 (um )k → 0,
as m → ∞,
(5.8)
with c given by (2.1). Applying Lemma 2.7, we may assume, without loss of generality, that um * u weakly in E. From this and (2.6), it is not difficult to show that u is a critical point of I. Thus, in order to prove Theorem A, it suffices to assume that u = 0.
H.F. Lins, E.A.B. Silva / Nonlinear Analysis 71 (2009) 2890–2905
In view of Proposition 4.4, it follows that 0 < α ≤ c < p∗ Lloc
1 N
(p−N )/p N /p
|K |∞
S
2903
. Hence Proposition 3.6 can be applied to show
that um → 0 strongly in (R ). Furthermore, by Proposition 3.7, there exist a sequence (ym ) ⊂ RN and r, η > 0 such that |ym | → ∞ as m → ∞, and
Z lim sup Br (ym )
m→∞
N
|um |p ≥ η > 0,
for all m ∈ N.
(5.9)
Without loss of generality we may assume that (ym ) ⊂ ZN . Then, defining vm (y) = um (y + ym ), m ∈ N, we have kvm k0 = kum k0 for all m ∈ N. Thus, taking a subsequence if necessary, there exists v ∈ E such that vm * v weakly in p E, vm → v strongly in Lloc (RN ) and vm (x) → v(x) almost everywhere in RN . From (5.9), we have v 6= 0. We claim that v is a critical point of I0 . Indeed, given w ∈ E, in view of (V ), (K ), (g1 ), (g2 ) and (g4 ), we have I00 (vm ).w → I00 (v).w,
as m → ∞.
(5.10)
On the other hand, considering wm (y) = w(y − ym ) for m ∈ N, in view of the periodicity of V0 , K0 and g0 , we get I00 (vm ).w = I00 (um ).wm ,
for all m ∈ N.
(5.11)
Moreover, applying Lemma 5.1, we have (5.12) |I00 (um ).wm − I 0 (um ).wm | → 0, as m → ∞. By kwm k0 = kwk0 for all m ∈ N, (5.8) and (5.12), we conclude that I00 (um ).wm → 0, as m → ∞. The above relation, (5.11) and (5.10) show that v is a critical point of I0 . The claim is proved. Our next task is to verify that I0 (v) ≤ c. In order to show this fact, we apply condition (K ) and the definition of (vm ) to get 1 0 1 I (um ).um ≥ p N
I (um ) −
Z
∗
+ p K0 (x)|vm | +
RN
Z
1 p
RN
g (x, um )um − G(x, um ) .
(5.13)
Now, in view of Fatou’s lemma, taking a subsequence if necessary, we may write lim inf
1
Z
N We claim that m→∞
RN
Z lim inf m→∞
1
∗
+ p K0 (x)|vm | ≥
1 p
RN
Z
N
∗
RN
K0 (x)|v + |p .
g (x, um )um − G(x, um ) ≥
Z RN
(5.14)
1 p
g0 (x, v)v − G0 (x, v) .
(5.15)
Assuming that the claim is true, we apply (5.8), (5.13)–(5.15) and the fact that (um ) ⊂ E is bounded to get c≥
1
Z
N
∗
RN
K0 (x)|v + |p +
Z RN
1 p
g0 (x, v)v − G0 (x, v) .
Since v is a critical point of I0 , it follows from the above relation that I0 (v) ≤ c. Now, in view of (g4 )-(i), (iii), and the definition of c, c ≤ max I (tu) ≤ max I0 (tu) = I0 (u) ≤ c . t ≥0
t ≥0
This implies that there exists γ ∈ Γ such that (2.3) holds. In view of Theorem 2.3, I possesses a critical point u ∈ Kc . From c ≥ α > 0 = I (0), it follows that u is a nontrivial critical point of I. Finally, we conclude the proof of Theorem A by showing that (5.15) holds. First we observe that, in view of Lemma 5.2,
Z RN
hi (x)|um |qi → 0,
as m → ∞, i = 2, 3,
(5.16)
since hi ∈ F , p ≤ qi < p∗ and um * 0 weakly in E. Invoking (5.16) and (g4 ), we get g (x, um )um − g0 (x, um )um → 0, strongly in L1 (RN ), as m → ∞. Similarly, G(x, um ) − G0 (x, um ) → 0, strongly in L1 (RN ), as m → ∞. Consequently, by the periodicity of g0 ,
Z lim inf m→∞
RN
1 p
g (x, um )um − G(x, um ) = lim inf m→∞
Z RN
1 p
g0 (x, vm )vm − G0 (x, vm ) .
Now, observe that, by (g3 ) and (g4 ), we have 1 1 g0 (x, s)s − G0 (x, s) ≥ −h1 (x) − h2 (x)|s|q2 − h3 (x)|s|q3 . p p Invoking (5.17), Lemma 5.2 and Fatou’s Lemma, we obtain (5.15). The proof of Theorem A is complete.
(5.17)
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5.2. Proof of Theorem B We argue as in the initial steps of the proof of Theorem A. Since g0 satisfies (g1 ) and (g2 ), applying Corollary 2.5, we may find a sequence (um ) ⊂ E such that I 0 ( um ) → c ≥ α > 0
and kI00 (um )k → 0,
as m → ∞,
(5.18)
where c is given by (2.1). By Lemma 2.7 we may assume, without loss of generality, that um * u weakly in E. From this and (2.6), it is not difficult to show that u is a critical point of I0 . Hence, in order to prove Theorem B, it suffices to assume that (p−N )/p N /p u = 0. Since, by Proposition 4.4, 0 < α ≤ c < N1 |K |∞ S , we may apply Proposition 3.6 to conclude that um → 0 p∗
strongly in Lloc (RN ). Furthermore, by Proposition 3.7, there exist a sequence (ym ) ⊂ RN and r, η > 0 such that |ym | → ∞, as m → ∞, and
Z lim sup m→∞
Br (ym )
|um |p ≥ η > 0,
for all m ∈ N.
(5.19)
As in the proof of Theorem A, we may assume that (ym ) ⊂ ZN . Then, defining vm (y) = um (y + ym ), m ∈ N, we have kvm k0 = kum k0 for every m ∈ N. Consequently, taking a subsequence if necessary, there exists v ∈ E such that vm * v p weakly in E, vm → v strongly in Lloc (RN ) and vm (x) → v(x) almost everywhere in RN . From (5.19), we have v 6= 0. It is not difficult to show that que v is a critical point of I0 . Indeed, given w ∈ E, by (V ), (K ), (g1 ) and (g2 ), we get I00 (vm ).w → I00 (v).w,
as m → ∞.
(5.20)
On the other hand, considering wm (y) = w(y − ym ), for m ∈ N, in view of the periodicities of V0 , K0 and g0 , we get I00 (vm ).w = I00 (um ).wm ,
for all m ∈ N.
Consequently, from (5.18) and the fact that kwm k0 = kwk0 for all m ∈ N, we conclude that I00 (um ).w → 0,
as m → ∞.
This limit together with (5.20) shows that v is a critical point of I0 as claimed. The Theorem B is proved.
Acknowledgements The first and the second authors were supported by CNPq/Brazil. References [1] H. Brézis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983) 437–477. [2] C.O. Alves, J.V. Gonçalves, Existence of positive solutions for m-Laplacian equations in RN involving critical Sobolev exponents, Nonlinear Anal. 32 (1998) 53–70. ∗ [3] A. Ambrosetti, M. Struwe, A note on the problem −∆u = λu + u|u|2 −2 , Manuscripta Math. 54 (1986) 373–379. [4] A. Capozzi, D. Fortunato, G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré, Anal. Non Linéaire 2 (1985) 463–470. [5] D.G. Costa, E.A.B. Silva, A note on problems involving critical Sobolev exponents, Differential Integral Equations 8 (1995) 673–679. [6] H. Egnell, Existence and nonexistence results for m-Laplace equations involving critical Sobolev exponents, Arch. Ration. Mech. Anal. 104 (1988) 57–77. [7] J. Garcia Azorero, I. Peral Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc. 323 (1991) 877–895. [8] F. Gazzola, B. Ruf, Lower-order perturbations of critical growth nonlinearities in semilinear elliptic equations, Adv. Differential Equations 2 (1997) 555–572. [9] J.C.N. Pádua, E.A.B. Silva, S.H.M. Soares, Positive solutions of critical semilinear problems involving a sublinear term at the origin, Indiana Univ. Math. J. 55 (3) (2006) 1091–1111. [10] E.A.B. Silva, S.H.M. Soares, Quasilinear Dirichlet problems in RN with critical growth, Nonlinear Anal. 43 (2001) 1–20. [11] E.A.B. Silva, M.S. Xavier, Multiplicity of solutions for quasilinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003) 341–358. [12] P.H. Rabinowitz, A note on semilinear elliptic equations on RN , in: A. Ambrosetti, A. Marino (Eds.), A Tribute in Honour of Giovanni Prodi, Quaderni, Scuola Normale Superiore, Pisa, 1991, pp. 307–318. [13] V. Coti Zelati, P.H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE in RN , Comm. Pure Appl. Math. 45 (1992) 1217–1269. [14] C. Troestler, M. Willem, Nontrivial solutions of a semilinear Schrödinger equation, Comm. Partial Differential Equations 21 (1996) 1431–1449. [15] W. Kryszewski, A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differential Equations 3 (1998) 441–472. [16] C.O. Alves, J.M. do Ó, O.H. Miyagaki, On perturbations of a class of a periodic m-Laplacian equation with critical growth, Nonlinear Anal. 45 (2001) 849–863. [17] C.O. Alves, P.C. Carrião, O.H. Miyagaki, Nonlinear perturbations of a periodic elliptic problem with critical growth, J. Math. Anal. Appl. 260 (2001) 133–146. [18] J. Chabrowski, Jianfu Yang, On Schrödinger equation with periodic potential and critical Sobolev exponent, Topol. Methods Nonlinear Anal. 12 (1998) 245–261. [19] J. Chabrowski, A. Szulkin, On a semilinear Schrödinger equation with critical Sobolev exponent, Proc. Amer. Math. Soc. 103 (2001) 85–93. [20] J.M. do Ó, B. Ruf, On a Schrödinger equation with periodic potential and critical growth in R2 , Nonlinear Differential Equations Appl. 13 (2006) 167–192.
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I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974) 324–353. G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1976) 353–372. A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381. P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, in: CBMS Regional Conf. Ser. in Math., vol. 65, AMS, Providence, RI, 1986. P.L. Lions, The concentration-compactness principle in the calculus of variations. The limit case part 1, 2, Rev. Mat. Iberoam. 1 (1985) 145–201. P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case II, Ann. Inst. H. Poincaré Anal. Non Lineaire 1 (1984) 223–283. C.O. Alves, J.M. do Ó, O.H. Miyagaki, Nontrivial solutions for a class of semilinear biharmonic problems involving critical exponents, Nonlinear Anal. 46 (2001) 121–133. J. Chabrowski, J.M. do Ó, On some fourth-order semilinear elliptic problems in RN , Nonlinear Anal. 49 (2002) 861–884. D.E. Edmunds, D. Fortunato, E. Jannelli, Critical exponents critical dimensions and the biharmonic operator, Arch. Ration. Mech. Anal. 112 (1990) 269–289.
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Quasilinear asymptotically periodic elliptic equations with critical growth Haendel F. Lins, Elves A.B. Silva ∗ Departamento de Matemática, Universidade de Brasília, 70910-900 Brasília-DF, Brazil
article
abstract
info
Article history: Received 19 November 2008 Accepted 21 January 2009
The existence of a nontrivial solution for a class of asymptotically periodic equations involving the p-Laplacian operator in the euclidean space is established. The solutions are obtained by an application of the Mountain Pass Theorem and the ConcentrationCompactness Principle. The lack of compactness due to the fact that the domain is unbounded is compensated by comparing the minimax level associated with the original problem and the minimax level associated with the periodic limit problem. © 2009 Elsevier Ltd. All rights reserved.
MSC: 35J20 35J60 Keywords: Quasilinear elliptic equations Sobolev critical exponent
1. Introduction In this article, we study the existence of a nontrivial solution for the quasilinear elliptic problem ∗
−∆p u + V (x)up−1 = K (x)up −1 + g (x, u), x ∈ RN , (1.1) u ∈ W 1,p (RN ), u ≥ 0, where ∆p u = div |∇ u|p−2 ∇ u is the p-Laplacian of u, 1 < p < N, p∗ = Np/(N − p) is the critical Sobolev exponent, and V , K : RN → R and g : RN × R → R are continuous functions.
Our main goal is to establish the existence of a nontrivial solution for the Problem (1.1) under an asymptotic periodicity condition at infinity. Considering F the class of functions h ∈ C (RN , R) ∩ L∞ (RN ) such that, for every ε > 0, the set {x ∈ RN : |h(x)| ≥ ε} has finite Lebesgue measure, we suppose that V and K are perturbations of periodic functions at infinity. More specifically, we assume that V and K satisfy
(V ) there exist a constant a0 > 0 and a function V0 ∈ C (RN , R), 1-periodic in xi , 1 ≤ i ≤ N, such that V0 − V ∈ F and V0 (x) ≥ V (x) ≥ a0 > 0, for all x ∈ RN ;
(K ) there exist a function K0 ∈ C (RN , R), 1-periodic in xi , 1 ≤ i ≤ N, and a point x0 ∈ RN , such that K − K0 ∈ F and (i) K (x) ≥ K0 (x) > 0, for all x ∈ RN , N −p (ii) K (x) = |K |∞ + O(|x − x0 | p−1 ), as x → x0 . Rs Considering G(x, s) = 0 g (x, t ) dt, the primitive of g, we also suppose the following hypotheses: (g1 ) g (x, s) = o(|s|p−1 ), as s → 0+ , uniformly in RN ; ∗
Corresponding author. Tel.: +55 61 3273 3356; fax: +55 61 3273 2737. E-mail addresses: [email protected] (H.F. Lins), [email protected] (E.A.B. Silva).
0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.171
H.F. Lins, E.A.B. Silva / Nonlinear Analysis 71 (2009) 2890–2905
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(g2 ) there are constants a1 , a2 > 0 and p < q1 < p∗ such that |g (x, s)| ≤ a1 + a2 |s|q1 −1 ,
for all (x, s) ∈ RN × [0, +∞);
(g3 ) there exist a constant p ≤ q2 < p∗ and functions h1 ∈ L1 (RN ), h2 ∈ F such that 1 p
g (x, s)s − G(x, s) ≥ −h1 (x) − h2 (x)|s|q2 ,
for all (x, s) ∈ RN × [0, +∞).
We observe that the conditions (g1 ) and (g2 ) allow us to employ variational methods to study Problem (1.1) and to verify that the associated functional has a local minimum at the origin. However, under these hypotheses, this functional does not satisfy the Palais–Smale compactness condition, since the domain is unbounded and the Problem (1.1) involves critical growth with respect to the Sobolev Imbedding. The condition (g2 ) imposes a subcritical growth on g. We also observe that the condition (g3 ) is used in the proof of the boundedness of the Palais–Smale sequences of the functional associated with (1.1). The asymptotic periodicity of g at infinity is given by the following condition:
(g4 ) there exist a constant p ≤R q3 < p∗ and functions h3 ∈ F , g0 ∈ C (RN × R, R), 1-periodic in xi , 1 ≤ i ≤ N, such that s (i) G(x, s) ≥ G0 (x, s) = 0 g0 (x, t ) dt, for all (x, s) ∈ RN × [0, +∞), (ii) |g (x, s) − g0 (x, s)| ≤ h3 (x)|s|q3 −1 , for all (x, s) ∈ RN × [0, +∞), ∗ (iii) the function (K0 (x)sp −1 + g0 (x, s))/sp−1 is nondecreasing in the variable s > 0. Finally, we also suppose that g satisfies:
(g5 ) there exists an open bounded set Ω ⊂ RN , containing x0 given by (K )-(ii), such that G(x,s) (i) sp → ∞, as s → ∞, uniformly in Ω , if N > p2 , G(x,s) (ii) sp log s → ∞, as s → ∞, uniformly in Ω , if N = p2 , (iii)
G(x,s)
p∗ −
s
p p−1
→ ∞, as s → ∞, uniformly in Ω , if N < p2 .
We observe that (g5 ) is essential to verify the fact that the mountain pass minimax level is in the interval where the ∗ local convergence in Lp (RN ) for the Palais–Smale sequences of the functional associated with the Problem (1.1) can be established. Now we may state our main result. Theorem A. Suppose that (V ), (K ) and (g1 )–(g5 ) are satisfied. Then Problem (1.1) has a nontrivial solution. The pioneering work on elliptic equations involving critical growth is due to Brézis and Nirenberg [1]. They studied the existence of a positive solution to a semilinear problem with critical growth in a bounded domain of RN , N ≥ 3. Since this famous article, related problems have been one of the main subjects of study in nonlinear analysis (see, for instance, [2–11] and the references therein). The existence of a nontrivial solution for periodic problems involving the Laplace operator and subcritical growth has been established by Rabinowitz [12]. Coti Zelati and Rabinowitz [13] proved that this class of problems has an infinite number of solutions. In the indefinite case (that is, the one in which the potential V can change sign), the existence of a nontrivial solution was established by Troestler and Willem [14] and Kryszewski and Szulkin [15]. The main characteristics of the problem studied in this article are the criticality of the growth and the unboundedness of the domain. Alves, do Ó and Miyagaki, in [16], studied Problem (1.1) with g (x, s) = h(x)sq−1 , where h ∈ C (RN , R) and V , K converging uniformly at infinity to periodic functions. Employing the Concentration-Compactness Principle, appropriate estimates on the mountain pass minimax level and comparison arguments, involving the Nehari manifold, between the minimax levels related with the Problem (1.1) and the periodic corresponding problem, these authors established the existence of nontrivial solutions for (1.1) and the corresponding periodic problem. We also observe that results for the Laplace operator, related to the ones obtained in [16], were established earlier by Alves, Carrião and Miyagaki [17] (see [27– 29] for related results). Finally, we observe that the existence of a nontrivial solution for a periodic problem involving the Laplace operator, critical growth and indefinite potential has been studied by Chabrowski and Yang [18], Chabrowski and Szulkin [19] and do Ó and Ruf [20]. In order to prove Theorem A, we adapt the arguments employed in [17,16]: we suppose, by contradiction, that the only solution for the Problem (1.1) is the trivial one. We use a version of the Mountain Pass Theorem without compactness condition [21] to get a Palais–Smale sequence associated with the minimax level. By appropriate estimates for this level, ∗ we establish, via the Concentration-Compactness Principle, the local convergence in Lp (RN ) of this Palais–Smale sequence. To obtain such estimates, adapting ideas contained in Brézis–Nirenberg [1] and Talenti [22], we use functions which are ∗ extremals for the Sobolev embedding D1,p (RN ) ⊂ Lp (RN ). Next, we use the obtained Palais–Smale sequence and a technical result due to Lions (cf. Coti Zelati and Rabinowitz [13]) to get a nontrivial critical point of the functional associated with the periodic problem. Furthermore, we are able to prove that the value of the functional associated with the Problem (1.1) at this point is less than or equal to the mountain pass minimax level and that this level is attained. We then employ a local
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version of the Mountain Pass Theorem (Theorem 2.3) to obtain a nontrivial critical point of the functional associated with the Problem (1.1). This contradicts our initial assumption that the only possible solution for Problem (1.1) is the trivial one. We observe that in our comparison argument we do not employ the Nehari manifold and, under the hypotheses of Theorem A, we do not know if the periodic problem associated with the Problem (1.1) has a nontrivial solution. However, in the particular case: V = V0 , g = g0 , K = K0 , Theorem A clearly gives us a nontrivial solution for the periodic problem. Indeed, in order to show the existence of a nontrivial solution for the periodic problem, condition (g4 ) is not necessary, that is, considering the problem
∗ −1
−∆p u + V0 (x)up−1 = K0 (x)up u ∈ W 1,p (RN ), u ≥ 0,
+ g0 (x, u),
x ∈ RN ,
(1.2)
under the hypotheses
(V0 ) the function V0 ∈ C (RN , R) is 1-periodic in xi , 1 ≤ i ≤ N, and there exists a constant a0 > 0 such that V0 (x) ≥ a0 > 0,
for all x ∈ RN ;
(K0 ) the function K0 ∈ C (R , R) is 1-periodic in xi , 1 ≤ i ≤ N, and there is a point x0 ∈ RN such that (i) K0 (x) > 0, for allx ∈ RN , N −p (ii) K0 (x) = kK0 k∞ + O(|x − x0 | p−1 ), as x → x0 ; N
and the function g0 satisfying (g1 )–(g3 ) and (g5 ), we may state: Theorem B. Suppose that (V0 ), (K0 ), (g1 )–(g3 ) and g5 are satisfied. Then Problem (1.2) possesses a nontrivial solution. 2. Preliminary results In this section we present two versions of the Mountain Pass Theorem due to Ambrosetti and Rabinowitz [23], which are essential tools in this article. We also introduce the variational framework associated with the Problem (1.1) and show the boundedness of the Palais–Smale sequences of the associated functional. 2.1. Versions of the Mountain Pass Theorem Let E be a real Banach space and I : E → R a C 1 functional. Let K be the set of critical points of I. Given b ∈ R, we define I b = {u ∈ E : I (u) ≤ b} and Kb = {u ∈ E : u ∈ K , I (u) = b}. The open ball in E with center at point x and radius r > 0 is denoted by Br (x). The symbol ∂ Br (x) denotes the boundary of this ball. Unfortunately, the functional associated with the Problem (1.1) does not satisfy the Palais–Smale condition. To overcome this difficulty, we use versions of the Mountain Pass Theorem. In the following, we state the first version of this Theorem (see also [23]). Theorem 2.1. Let E be a real Banach space. Suppose I ∈ C 1 (E , R) is such that I (0) = 0 and
(I1 ) there exist ρ, α > 0 such that I |∂ Bρ (0) ≥ α > 0, (I2 ) there exists e ∈ E \Bρ (0) such that I (e) ≤ 0. Then I possesses a (PS )c sequence for c ≥ α > 0 given by c = inf max I (γ (t )),
(2.1)
Γ = {γ ∈ C ([0, 1], E ) : γ (0) = 0, γ (1) ∈ I 0 ∩ (E \Bρ (0))}.
(2.2)
γ ∈Γ t ∈[0,1]
where
Proof. Supposing by contradiction that the theorem is false, it follows that the condition (PS )c is vacuously satisfied. By the Mountain Pass Theorem [24], there exists a critical point u ∈ E on level c, that is, I (u) = c and I 0 (u) = 0. Then the sequence (um ) given by um = u for all m ∈ N is a (PS )c sequence for I. This is a contradiction, and the proof of Theorem 2.1 is complete. We will also need to establish a local version of Theorem 2.1. In order to show this fact, we first state a local deformation result, given by the lemma below. Lemma 2.2. Let E be a real Banach space. Assume that I ∈ C 1 (E , R) and, for some b ∈ R, there exists a compact set D ⊂ I b such that D 6= ∅ and D ∩ Kb = ∅. Then, given ε¯ > 0, there exist 0 < ε < ε¯ and η ∈ C ([0, 1] × E , E ) such that
(η1 ) η(t , u) = u, for all t ∈ [0, 1], I (u) 6∈ [b − ε¯ , b + ε¯ ], (η2 ) I (η(t , u)) ≤ I (u) for all u ∈ E and t ∈ [0, 1], (η3 ) η(1, D) ⊂ I b−ε .
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Theorem 2.3. Let E be a real Banach space. Suppose that I ∈ C 1 (E , R) satisfies I (0) = 0, (I1 ) and (I2 ). If there exists γ0 ∈ Γ , Γ defined by (2.2), such that c = max I (γ0 (t )) > 0,
(2.3)
t ∈[0,1]
then I possesses a nontrivial critical point u ∈ Kc ∩ γ0 ([0, 1]). Proof. Arguing by contradiction, suppose that γ0 ([0, 1]) ∩ Kc = ∅. Thus, let D = γ0 ([0, 1]) and 0 < ε¯ < c and apply Lemma 2.2 to find 0 < ε < ε¯ and η ∈ C ([0, 1] × E , E ) satisfying conditions (η1 )–(η3 ). Taking γ (t ) = η(1, γ0 (t )) for all t ∈ [0, 1], it follows from I (0) = 0, (I2 ), (η1 ) and our choice of ε¯ that γ ∈ Γ . Furthermore, by (2.3) and (η3 ), max I (γ (t )) ≤ c − ε.
t ∈[0,1]
However, this inequality contradicts the definition of c. The proof of the theorem is concluded.
2.2. The variational framework Consider the Sobolev space E = W 1,p (RN ) endowed with one of the following norms p
Z
|∇ u|p + V (x)|u|p ,
k uk = RN
kukp0 =
Z
|∇ u|p + V0 (x)|u|p .
RN
Note that, in view of (V ), the above norms are both equivalent to the standard norm on E. The functional I : E → R associated with the Problem (1.1) is given by I (u) =
1 p
k uk p −
Z
1 p∗
∗
RN
K (x)|u+ |p −
Z RN
G(x, u),
(2.4)
where, in order to obtain a nonnegative solution, we set g (x, s) = 0, for x ∈ RN , s < 0. Under the hypotheses (V ), (K ), (g1 ) and (g2 ) we have that I ∈ C 1 (E , R). Furthermore, the critical points of this functional are the weak solutions for the Problem (1.1). In a similar fashion, the functional I0 ∈ C 1 (E , R), defined by I0 (u) =
1 p
p 0
k uk −
1
Z
p∗
+ p∗
RN
K0 (x)|u |
Z − RN
G0 (x, u),
(2.5)
is associated with the periodic problem. We also suppose that g0 (x, s) = 0, for x ∈ RN , s < 0. Observe that, given δ > 0, in view of (g1 ) and (g2 ), we may find a constant Cδ > 0 such that
|g (x, s)| ≤ δ|s|p−1 + Cδ |s|q1 −1 , for all (x, s) ∈ RN × R, δ Cδ |G(x, s)| ≤ |s|p + |s|q1 , for all (x, s) ∈ RN × R. p
q1
(2.6) (2.7)
These estimates and a standard argument (see [24]) show that the functional I satisfies the geometric conditions (I1 ) and (I2 ). Since the proof of this fact is standard, we just state it: Lemma 2.4. Suppose that (V ), (K ), (g1 ) and (g2 ) are satisfied. Then the functional I defined by (2.4) satisfies I (0) = 0, (I1 ) and (I2 ). As a consequence of Theorem 2.1 and Lemma 2.4, we have Corollary 2.5. Suppose that (V ), (K ), (g1 ) and (g2 ) are satisfied. Then the functional I possesses a (PS )c sequence, with c given by (2.1). As a final result of this section, we verify the boundedness of the (PS ) sequences associated to the functional I. First, however, we establish a simple result that will be employed several times in our work. From now on, |E | will denote the Lebesgue measure of a measurable set E ⊂ RN . In the following lemma, given h ∈ F , we set Dε = {x ∈ RN : |h(x)| ≥ ε} and Dε (R) = {x ∈ RN : |h(x)| ≥ ε, |x| ≥ R}. Lemma 2.6. Suppose h ∈ F . Then |Dε (R)| → 0 as R → ∞, Proof. Since h ∈ F , |Dε | < ∞ for all ε > 0. In order to prove the lemma, it suffices to verify the following claim: lim |Dε ∩ (RN \BRm )| = 0
m→∞
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for each sequence (Rm ) ⊂ R such that Rm → ∞. Consider the real function f : RN → R given by f (x) = χDε (x), that is, f (x) =
for x ∈ Dε for x 6∈ Dε .
1 0
Then f ∈ L1 (RN ) and |f |1 = RN |f | = |Dε |. Moreover, defining the sequence of functions fm : RN → R by fm (x) = χDε ∩(RN \BRm ) (x), it follows that |fm | ≤ |f |. Since fm → 0 almost everywhere in RN as m → ∞, our claim follows from Lebesgue’s Dominated Convergence Theorem.
R
Lemma 2.7. Suppose that (V ), (K ), (g1 ), (g2 ) and (g3 ) are satisfied. Then every (PS ) sequence is bounded. Proof. Let (um ) ⊂ E be a (PS )b sequence, with b ∈ R. Arguing by contradiction, we suppose that (um ) is not bounded. Without loss of generality, kum k → ∞ as m → ∞. First, we claim that there exists λ2 > 0 such that, for m sufficiently large, we have ∗
kum kp ≤ λ2 |um |pp∗ .
(2.8)
Indeed, since I (um ) → 0 strongly in E , we may write 0
∗
k um k p =
Z RN
Z
g (x, um )um +
∗
RN
K (x)|um |p + o(kum k).
(2.9)
Given δ > 0, we invoke (2.6), (2.9) and the Sobolev Embedding Theorem to find C1 > 0 and C2 = C2 (δ) > 0 such that ∗
kum kp ≤ δ C1 kum kp + C2 |um |qq11 + |K |∞ |um |pp∗ + o(kum k). Thus, since p > 1, taking δ > 0 sufficiently small and m sufficiently large, we obtain ∗
kum kp ≤ 2C2 |um |qq11 + 2|K |∞ |um |pp∗ .
(2.10)
Now, setting 0 < β < 1 such that q1 = β p + (1 − β)p , we apply Young’s inequality to find ∗
∗
|um |qq11 ≤ βε1/β |um |pp + (1 − β)ε −1/(1−β) |um |pp∗ ,
(2.11)
for all ε > 0. Choosing ε > 0 sufficiently small, we apply (2.10), (2.11) and the Sobolev Embedding Theorem to find λ2 > 0 such that (2.8) holds. The claim is proved. Now, invoking (g3 ), we have I ( um ) −
1 0 1 I (um ).um = p N
≥
1
Z
∗
N
ZR
N
RN
K (x)|um |p +
Z RN
1 p
g (x, um )um − G(x, um )
p∗
K (x)|um | dx − |h1 |1 −
Z RN
h2 (x)|um |q2 .
Thus, again using the fact that (um ) is a (PS )b sequence, we find C3 > 0 such that 1 N
Z
∗
RN
K (x)|um |p ≤ C3 +
Z RN
h2 (x)|um |q2 + o(kum k).
(2.12)
Given ε > 0, we set Dε (R) = {x ∈ RN : |x| ≥ R, |h2 (x)| ≥ ε}, for all R > 0. Then, since h2 ∈ F , applying Lemma 2.6, we find R = Rε > 0 such that |Dε (R)| < ε . By Hölder’s inequality,
Z Dε (R)
∗ ∗ q h2 |um |q2 ≤ |h2 |∞ ε (p −q2 )/p |um |p2∗
(2.13)
and
Z RN \D
ε (R)
h2 |um |q2 dx ≤ ε|um |qq22 + |h2 |∞
wN RN
(p∗ −q2 )/p∗
N
q
|um |p2∗ .
Furthermore, considering 0 < γ ≤ 1 such that q2 = γ p + (1 − γ )p∗ , we may invoke Hölder’s inequality, the Sobolev p∗
q
Embedding Theorem and (2.8) to find C4 > 0 such that |um |q22 ≤ C4 |um |p∗ . Consequently,
Z
q2
RN \D
ε (R)
h2 |um |
p∗ C4 um p∗
≤ε |
| + |h2 |∞
wN RN N
(p∗ −q2 )/p∗
q
|um |p2∗ .
(2.14)
Choosing ε > 0 sufficiently small, we use (2.8), (2.12)–(2.14), condition (K ) and the fact that q2 < p∗ to conclude that (um ) ∗ is bounded in Lp (RN ). Furthermore, using (2.8) one more time, it follows immediately that (um ) is also bounded in E. The proof of Lemma 2.7 is complete.
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3. Concentration compactness In this section we state the Concentration-Compactness Principle due to Lions [25], which is fundamental to overcome the lack of compactness caused by the presence of a critical term in the Problem (1.1), and establish some related results. We also present results concerning the behaviour of the Palais–Smale sequences: we pointed out the strong convergence in p−N p
p∗
N
Lloc (RN ) of (PS )b sequences when b < N1 |K |∞ S p and a proposition which will be essential to guarantee that the solution that we provide in our proof of Theorem A is nontrivial. 3.1. The Concentration-Compactness Principle We first recall that the best constant for the Sobolev embedding D1,p (RN ) ⊂ Lp (RN ) is given by ∗
R S=
inf
u∈D1,p (RN )
RN
u6=0
|∇ u|p p/p∗ . |u|p∗
RN
R
(3.1)
Now we state the Concentration-Compactness Principle due to Lions [25]. Lemma 3.1. Suppose 1 ≤ p < N and let (um ) ⊂ D1,p (RN ) be a sequence such that um * u weakly in D1,p (RN ) and ∗ |∇ um |p dx * µ, |um |p dx * ν weakly in the sense of measures, where µ and ν are nonnegative bounded measures in RN . Then there exist a set at most countable J and a family {xj : j ∈ J } of points in RN such that ∗
(a) ν = |u|p dx + j∈J νj δxj , P (b) µ ≥ |∇ u|p dx + j∈J µj δxj ,
P
p/p∗
where {νj : j ∈ J } and {µj : j ∈ J } are two families of positive numbers such that S νj
P
j∈J
p/p∗ j
ν
≤ µj for all j ∈ J. In particular,
< ∞. ∗
p P Given K by condition (K ) and the measure ν established by Lemma 3.1, we denote by K ν the measure K |u| dx + K ( x )ν δ , then we state some auxiliary results. j j x j∈J j
Lemma 3.2. Suppose that (V ), (K ), (g1 ) and (g2 ) are satisfied. If um * u weakly in E, where xj , νj , j ∈ J are given by Lemma 3.1, we have: V |um |p → V |u|p in L1loc (RN ), g (x, um )um → g (x, u)u in
L1loc
(3.2)
(R ), N
(3.3)
p∗
K |um | dx * K ν weakly in the sense of measures.
(3.4)
In the following, we refer the reader to the references [2] and [7]. We fix ϕ ∈ C0∞ (RN , [0, 1]), ϕ ≡ 1 in B1/2 (0), ϕ ≡ 0 in RN \B1 (0). Given ε > 0, j ∈ J and xj ∈ RN as in Lemma 3.1, consider ϕε = ϕεj : RN → R defined by
ϕε (x) = ϕ
x − xj
ε
,
for every x ∈ RN .
(3.5)
Lemma 3.3. Suppose that (K ) is satisfied. Let µ and ν be as in Lemma 3.1 and 1 ≤ q < ∞. Then we have
Z RN
ϕεq dν → νj ,
Z RN
ϕεq dµ → µj ,
Z RN
K ϕεq dν → K (xj )νj ,
as ε → 0.
Lemma 3.4. Suppose that (V ), (K ), (g1 ) and (g2 ) are satisfied. Let (um ) ⊂ E be a bounded sequence satisfying um * 0 weakly in E, I 0 (um ) → 0 strongly in E ∗ , as m → ∞. Then, letting xj , νj , j ∈ J be as in Lemma 3.1, we have J = ∅ or νj ≥ all j ∈ J .
S K (xj )
N /p
3.2. Behaviour of the Palais–Smale sequences We start with the following auxiliary result: Lemma 3.5. Suppose that (g1 )–(g3 ) are satisfied. Let (um ) ⊂ E be a bounded sequence such that um * 0 weakly in E. Then
Z lim inf m→∞
RN
1 p
g (x, um )um − G(x, um ) ≥ 0.
for
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Proof. Arguing by contradiction, we suppose that there exist a subsequence, still denoted by (um ), a > 0 and m1 ∈ N such that
Z
1 p
RN
g (x, um )um − G(x, um ) ≤ −a < 0,
(3.6)
for all m ≥ m1 . Now, by the condition (g3 ),
Z
1 p
RN \BR (0)
Z
g (x, um )um − G(x, um )
Z
dx ≥ − RN \BR (0)
h1 −
RN \BR (0)
h2 |um |q2 ,
(3.7)
for all R > 0, with q2 ∈ (p, p∗ ). Given η > 0, since h1 ∈ L1 (RN ), there exists R0 > 0 such that, for R ≥ R0 ,
Z
|h1 | <
RN \BR (0)
η 4
.
(3.8)
Taking δ > 0 to be chosen later, define Dδ (R) = {x ∈ RN : |h2 (x)| ≥ δ, |x| ≥ R}. Since h2 ∈ F , we may choose R1 > 0 such that |Dδ (R)| < δ for all R ≥ R1 . But,
Z RN \BR (0)
h2 |um |q2 =
Z Dδ (R)
h2 |um |q2 +
Z δ (R)∪BR (0))
RN \(D
h2 |um |q2 ,
since Dδ (R) ⊂ R \BR (0). In view of Hölder’s inequality and our choice of R1 , we have, for all R ≥ R1 , N
Z
q2
Dδ (R)
|h2 ||um |
Z ≤ |h2 |∞
p∗
Dδ (R)
q∗2 p
|um |
|Dδ (R)|(p
∗ −q )/p∗ 2
≤ c1 δ (p
∗ −q )/p∗ 2
,
and
Z RN \(Dδ (R)∪BR (0))
|h2 ||um |q2 ≤ δ
Z RN
|um |q2 ≤ c2 δ,
where c1 and c2 are positive constants, obtained via the Sobolev Embedding Theorem and the boundedness of (um ) ⊂ E. Thus
Z RN \BR (0)
|h2 ||um |q2 ≤ c1 δ
p∗ −q2 p∗
+ c2 δ.
Therefore, we may choose δ > 0 such that, for R ≥ R1 > 0,
Z RN \BR (0)
|h2 ||um |q2 <
η 4
.
This inequality, (3.7) and (3.8), give us
Z RN \BR (0)
1 p
η g (x, um )um − G(x, um ) ≥ − ,
(3.9)
2
for every R ≥ max{R0 , R1 }. Now, for such R, from (g1 ) and (g2 ), there exists c3 > 0 such that
Z
BR (0)
Z g (x, um )um − G(x, um ) ≤ p
1
BR (0)
|um |p + c3 |um |q1 ,
where q1 ∈ (p, p ) is given by (g2 ). Since um * 0 weakly in E, taking a subsequence if necessary, we may suppose that q1 um → 0 in Lloc (RN ). Thus there exists m0 = m0 (η) ∈ N such that, for m ≥ m0 , ∗
Z BR (0)
1
η
g (x, um )um − G(x, um ) ≥ − . p 2
(3.10)
By our choice of R, (3.9) and (3.10), we have
Z RN
1 p
g (x, um )um − G(x, um ) ≥ −η,
for all m ≥ m0 . This contradicts (3.6), since η < a, and concludes the proof of the lemma.
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Proposition 3.6. Suppose that (V ), (K ) and (g1 )–(g3 ) are satisfied. Let (um ) ⊂ E be a (PS )b sequence with 0 < b < 1 N
(p−N )/p N /p
|K |∞
p∗
and um * 0 weakly in E. Then, up to a subsequence, um → 0 strongly in Lloc (RN ) as m → ∞.
S
Proof. In view of Lemma 3.1, taking a subsequence if necessary, we have ∗
|um |p dx * ν =
X
νj δxj ,
j∈J
|∇ um |p dx * µ ≥
X
µj δ x j ,
j∈J
weakly in the sense of measures, where J is a set at most countable and νj , µj > 0 for every j ∈ J . We claim that J = ∅. Arguing by contradiction, suppose that there exists j0 ∈ J. Since I (um ) → b < I 0 (um ) → 0 and (um ) ⊂ E is bounded, we find m1 ∈ N such that p−N
1
p−N
1 N
N
|K |∞p S p ,
1
N
|K |∞p S p > I (um ) − I 0 (um ).um N p Z Z 1 1 ∗ K |um |p + = g (x, um )um − G(x, um ) , N
RN
p
RN
for all m ≥ m1 . In view of the Lemma 3.5, taking a greater m1 if necessary, we may assume that 1 N
p−N
N
1
1
p
N
|K |∞p S p > I (um ) − I 0 (um ).um ≥
Z
∗
RN
K |um |p ,
(3.11)
for all m ≥ m1 . Now, taking ϕ ∈ C0∞ (RN , [0, 1]) such that ϕ(xj0 ) = 1 and invoking (3.11), it follows that 1 N
p−N
1
N
|K |∞p S p >
N
Z
∗
RN
ϕ K |um |p dx,
for all m ≥ m1 . Letting m → ∞, in view of (3.11) and Lemmas 3.2 and 3.4 we get 1 N
p−N
1 X
N
|K |∞p S p >
N j∈J
1
K (xj )νj ϕ(xj ) ≥
N
K (xj0 )νj0 ≥
1 N
K (xj0 )
p−N p
N
Sp.
This contradicts K (xj0 ) ≤ |K |∞ . The claim is proved. ∗
Then, we may conclude that |um |p dx * 0 weakly in the sense of measures in RN . Indeed, given a compact set L ⊂ RN , we choose ϕ ∈ C0∞ (RN , [0, 1]) such that ϕ ≡ 1 em L to get
Z
∗
|um |p ≤
0≤
Z
∗
RN
L
|um |p ϕ → 0,
as m → ∞.
This concludes the proof of the proposition.
p−N p
N
Proposition 3.7. Suppose that (V ), (K ), (g1 ) and (g2 ) are satisfied. Let (um ) ⊂ E be a (PS )b sequence with 0 < b < N1 |K |∞ S p , and um * 0 weakly in E as m → ∞. Then there exist a sequence (ym ) ⊂ RN and r , η > 0 such that |ym | → ∞ and
Z lim sup Br (ym )
m→∞
|um |p ≥ η > 0.
Proof. Supposing that the result does not hold, we have (see [13] or [26]):
Z RN
|um |σ → 0,
for every σ ∈ (p, p∗ ).
(3.12)
Since (um ) ⊂ E is bounded and I 0 (um ) → 0 in E ∗ , we get
k um k p −
Z RN
g (x, um )um −
Z
∗
RN
K |um |p = om (1).
(3.13)
We claim that
Z lim
m→∞
RN
g (x, um )um = 0.
(3.14)
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Indeed, from (2.6), (3.12) and the fact that (um ) ⊂ E is bounded, we find C2 > 0 such that
Z
|g (x, um )um | ≤ δ C2 .
lim sup RN
m→∞
Since δ > 0 can be chosen sufficiently small, the above inequality establishes (3.14), proving the claim. Similarly, we conclude that
Z lim
m→∞
RN
G(x, um ) = 0.
(3.15)
Since b > 0, we may not have um → 0 strongly in E. Thus, taking a subsequence if necessary, kum kp → l > 0 as m → ∞. From (3.13) and (3.14), we get
Z
∗
RN
K |um |p → l. p−N
Since I (um ) → b < b=
1
(3.16)
1 N
N
|K |∞p S p , in view of (3.15) and (3.16), we obtain
l.
N
(3.17)
But, from the definition of S,
Z
p
k um k ≥
Z
S
p
|∇ um | ≥
K |um |
p
∗
RN
p∗
RN
p |K |∞
p p∗
.
Letting m → ∞ and using (3.16) and (3.17), we get S
bN ≥
(bN )p/p . ∗
p/p∗
|K |∞ Consequently, b≥
1 N
|K |(∞p−N )/p S N /p ,
contradicting the fact that b <
1 N
(p−N )/p N /p
|K |∞
S
. The proposition is proved.
4. Estimates In this section we shall verify that the minimax level associated with the Mountain Pass Theorem (Theorem 2.1) is in ∗ the interval where there is local convergence in Lp (RN ) of the Palais–Smale sequences for the functional associated with (1.1) and the Proposition 3.7 can be applied. To show this result, we use test functions as the ones employed by Brézis and Nirenberg [1], and we verify some auxiliary results about these functions. Then we show the main result of the section, providing the estimate for the minimax level c given by (2.1). 4.1. Test functions Without loss of generality, we assume that x0 , given by the condition (K ), is the origin of RN and that B2 = B2 (0) ⊂ Ω , with Ω given by the condition (g5 ). Given ε > 0, we consider the function wε : RN → R defined by
ε (N −p)/p , (ε + |x|p/(p−1) )(N −p)/p 2
wε (x) = C (N , p) where C (N , p) =
" N
N −p p−1
p−1 #(N −p)/p2
.
We observe that {wε }ε>0 is a family of functions on which the infimum that defines the best constant, S, for the Sobolev ∗ embedding D1,p (RN ) ⊂ Lp (RN ) is attained. We also consider φ ∈ C0∞ (RN , [0, 1]), φ ≡ 1 in B1 , φ ≡ 0 in RN \B2 and define uε uε = φwε , vε = R . ∗ ( RN Kupε )1/p∗
H.F. Lins, E.A.B. Silva / Nonlinear Analysis 71 (2009) 2890–2905
2899
The following lemmas were inspired by [2,8]. We just state them, since their proofs are standard. Lemma 4.1. Suppose that (K ) is satisfied. Then, there exist positive constants k1 , k2 and ε0 such that
Z RN \B1
k1 <
|∇ uε |p = O(ε (N −p)/p ),
Z
∗
RN
Z
Kupε < k2 ,
as ε → 0+ ,
for all 0 < ε < ε0 ,
|x|(N −p)/(p−1) wεp = O(ε(N −p)/p ), ∗
|x|≤1
Z RN
(4.1)
(4.2)
as ε → 0+ ,
|∇vε |p ≤ |K |(∞p−N )/N S + O(ε(N −p)/p ),
(4.3)
as ε → 0+ .
(4.4)
Lemma 4.2. Suppose that (K ) is satisfied. Then, as ε → 0, we have N −p
p
(i) |vε |p = O(ε p ), if N < p2 ; p (ii) |vε |p = O ε p−1 | log ε| , if N = p2 ; p
(iii) |vε |p = O ε p−1 , if N > p2 .
4.2. Estimates for the minimax level Before stating the main result of this section, we present a useful lemma. Lemma 4.3. Suppose that (V ), (K ), (g1 ) and (g2 ) are satisfied. Consider tε > 0 such that I (tε vε ) = maxt ≥0 I (t vε ). Then, there exist ε0 > 0 and positive constants T1 and T2 such that T1 ≤ tε ≤ T2 for every 0 < ε < ε0 . Proof. First, we observe that, in view of Lemma 2.4, I satisfies I (0) = 0 and (I1 ). Consequently, since I ∈ C 1 (E , R) and kvε k is bounded, we conclude that there exists T1 > 0 such that tε ≥ T1 > 0 for every R ε > ∗0 sufficiently small. On the other hand, the estimate tε ≤ T2 < ∞ is obtained using the definition of I, the fact that RN K vεp = 1, (2.7) and the boundedness of kvε k. This proves the lemma. Next we state a result which provides an appropriate estimate on the minimax level. Proposition 4.4. Suppose that (V ), (K ), (g1 ), (g2 ) and (g5 ) are satisfied. Then there exists v ∈ E \{0} such that max I (t v) < t ≥0
1 N
p−N
N
|K |∞p S p .
Proof. Consider tε as defined by Lemma 4.3. Letting Xε to be the integral
R
RN
|∇vε |p , we get
∗
tp tp I (tε vε ) ≤ ε (Xε + |V |∞ |vε |pp ) − G(x, tε vε ) − ε∗ N p p Z R 1 N /p p ≤ Xε + c1 |vε |p − G(x, tε vε ), N RN
Z
for some constant c1 > 0. Indeed, considering the function h = hε : [0, ∞) → R given by h(t ) = 1/(p∗ −p)
that t0 = Xε
I (tε vε ) ≤
1 N
is a maximum point of h and h(t0 ) = p−N
N
(|K |∞N S + O(ε (N −p)/p )) p + c1 |vε |pp −
1 N /p X . N ε
Z RN
1 X tp p ε
−
1 p∗ t , p∗
we have
It follows from (4.4) that
G(x, tε vε ).
Applying the inequality
(b + c )ζ ≤ bζ + ζ (b + c )ζ −1 c ,
b, c ≥ 0, ζ ≥ 1,
we get I (tε vε ) ≤
1 N
|K |(∞p−N )/p S N /p + c1 |vε |pp −
Z RN
G(x, tε vε ) + O(ε
N −p p
).
(4.5)
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Now consider
N −p ε p , γ (ε) = ε p−1 | log ε|, p−1 ε ,
if N < p2 , if N = p2 , if N > p2 .
(4.6)
In view of Lemma 4.2, (4.5) and (4.6), we find a constant M > 0 such that I (tε vε ) ≤
1 N
|K |(∞p−N )/p S N /p + γ (ε) M −
Z
1
γ (ε)
RN
G(x, tε vε ) .
(4.7)
In order to prove Proposition 4.4, we just need to verify that lim
ε→0+
Z
1
γ (ε)
RN
G(x, tε vε ) = +∞.
(4.8)
By (g1 ) and (g5 ), it is possible to choose c > 0 such that G(x, s) +
1 p
csp ≥ 0,
for every x ∈ Ω , s ≥ 0.
(4.9)
Given A0 > 0, we invoke (g5 ) again to obtain R = R(A0 ) > 0 such that, for x ∈ Ω , s ≥ R,
G(x, s) ≥
1 A0 sµ , p 1 p 1 p
if N < p2 ,
A0 sp log s,
if N = p2 ,
A0 sp ,
if N > p2 ,
(4.10)
where µ = p∗ − p/(p − 1). Now consider the function ηε : [0, ∞) → R defined by
ε (N −p)/p . (ε + r p/(p−1) )(N −p)/p 2
ηε (r ) =
Since φ ≡ 1 in B1 , in view of (4.2), we find a constant c1 > 0 such that vε (x) ≥ c1 ηε (|x|) for |x| < 1. Furthermore, since ηε is decreasing, there exists a positive constant α˜ such that, for |x| < ε tε vε (x) ≥ T1 c1 ηε (|x|) ≥ T1 c1 ηε (ε
p−1 p
) ≥ αε ˜
N −p (1−p) p2
p−1 p
,
.
Here T1 is given by Lemma 4.3. Then we may choose ε1 > 0 such that tε vε (x) ≥ αε ˜ for |x| < ε
p−1 p
N −p (1−p) p2
≥ R = R(A0 ),
(4.11)
, 0 < ε < ε1 . It follows from (4.10) and (4.11) that
1 A0 tεµ vεµ , p 1 A0 tεp vεp log(tε vε ), G(x, tε vε (x)) ≥ p 1 A0 tεp vεp , p
if N < p2 , if N = p2 ,
(4.12)
if N > p2 ,
p−1
for |x| < ε p , 0 < ε < ε1 . Now, in order to verify (4.8), we should consider the three possibilities given by (g5 ). However, since they are similar, we will just consider the case N < p2 . Since B2 (0) ⊂ Ω , we invoke (4.9) and (4.12), to get
Z RN
G(x, tε vε ) =
Z
G(x, tε vε ) + B (p−1)/p ε
≥
1 p
A0 tεµ
Z
Z Ω \B (p−1)/p ε
c
B (p−1)/p ε
vεµ − tεp p
G(x, tε vε )
Z Ω \B (p−1)/p ε
vεp
H.F. Lins, E.A.B. Silva / Nonlinear Analysis 71 (2009) 2890–2905
≥
1 p
Z
µ
2901
1
A0 T1
B (p−1)/p ε
vεµ − cT2p |vε |pp , p
for 0 < ε < ε1 , where T1 and T2 are given by Lemma 4.3. Consequently, by Lemma 4.2, there exists M1 > 0 such that
Z RN
G(x, tε vε ) ≥
1 p
Z
µ
A0 T1
B (p−1)/p ε
vεµ − M1 ε
Now, considering the change of variable s = ε
Z B (p−1)/p ε
vεµ (x) dx
µ
Z
≥ c1
B (p−1)/p ε
1−p p
N −p p
µ
(4.13)
= c1 ωN
ε
Z
p−1 p
ε
N −p µ N −1 p2
r
N −p p µ dr p p − 1 ε+r
0
s
N −1
ds ε
p
1 + s p−1
= c2 ε
.
t,
ηεµ (|x|) dx
Z 1 µ = c1 ωN 0
N −p p
N −p p µ
(N −p)(p−1) p2
(p∗ −µ)
,
since µ = p − p/(p − 1). Thus, by (4.13), ∗
1
ε
N −p p
Z RN
G(x, tε vε ) ≥
c2 µ T p 1
A0 − M1 .
Since A0 > 0 can be chosen arbitrarily large, the above relation establishes (4.8). The Proposition 4.4 is proved.
5. Proofs of Theorems A and B In this section we prove Theorems A and B by verifying that the functionals I and I0 , defined by (2.4) and (2.5), respectively, have nontrivial critical points. 5.1. Proof of Theorem A Before proving Theorem A, we introduce two technical results. Lemma 5.1. Suppose that (V ), (K ) and (g4 ) are satisfied. Let (um ) ⊂ E be a bounded sequence and wm (y) = w(y − ym ), where w ∈ W 1,p (RN ) and (ym ) ⊂ RN . If |ym | → ∞, then we have ∗
(K0 − K )|um |p −1 wm → 0, (V0 − V )|um |p−1 wm → 0, (g0 (x, um ) − g (x, um ))wm → 0, strongly in L1 (RN ), as m → ∞. Proof. Considering that the other two limits follow by a similar argument, we will establish the first limit in Lemma 5.1. ∗ Given δ > 0, since w ∈ Lp (RN ), we find 0 < ε < δ such that, for every measurable A ⊂ RN satisfying |A| < ε ,
Z
∗
|w|p < δ.
(5.1)
A
We fix ε > 0 and set Dε (R) = {x ∈ RN : |K (x) − K0 (x)| ≥ ε, |x| ≥ R}. By the condition (K ) and the Lemma 2.6, we may find R > 0 such that |Dε (R)| < ε . Then, by Hölder’s inequality, condition (K ), (5.1) and the fact that (um ) ⊂ E is bounded, we obtain a constant C1 > 0 such that
Z
∗ −1
RN \BR (0)
p |K (x) − K0 (x)||u+ m|
∗
1 |wm | ≤ 2kK k∞ |um |pp∗− |wm |Lp∗ (Dε (R)) + ε|um |pp∗ −1 |w|p∗ ∗ 1
≤ C1 (δ p∗ + δ).
(5.2)
On the other hand, by Hölder’s inequality, the condition (K ) and the fact that (um ) ⊂ E is bounded, we find C2 > 0 such that
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Z
∗ −1
BR (0)
p |K (x) − K0 (x)||u+ m|
∗
|wm | ≤ 2kK k∞ |um |pp∗ −1 |w(. − ym )|Lp∗ (BR (0)) Z ≤ C2
∗
BR (−ym )
|w(y)|p
1/p∗
.
∗
Thus, since w ∈ Lp (RN ) and |ym | → ∞, there exists m0 ∈ N such that
Z
∗ −1
BR (0)
p |K (x) − K0 (x)||u+ m|
|wm | ≤ C2 δ,
for all m ≥ m0 .
(5.3)
p∗ −1 The inequalities (5.2), (5.3) and the fact that δ > 0 can be chosen arbitrarily small imply that (K (x)− K0 (x))|u+ wm → 0 m| 1 N strongly in L (R ) as m → ∞. The proof of Lemma 5.1 is complete.
In order to prove Theorem A we also need of the following result. Lemma 5.2. Suppose p ≤ q < p∗ and h ∈ F . Let (um ) ⊂ E be a sequence such that um * u weakly in E as m → ∞. Then h(x)|um |q → h(x)|u|q strongly in L1 (RN ),
as m → ∞.
Proof. Arguing by contradiction, we suppose that there exist a subsequence, still denoted by (um ), and ε > 0 such that
Z RN
|h(x)| · | |um |q −|u |q | ≥ ε > 0,
for all m ∈ N.
(5.4)
Now we define Dδ (R) = {x ∈ RN : |h(x)| ≥ δ, |x| ≥ R}. Since h ∈ F , there exists R = Rδ > 0 such that |Dδ (R)| < δ . Consequently, applying Hölder’s inequality, we get
Z Dδ (R)
|h(x)| · | |um |q −|u |q | ≤ |h|∞
Z Dδ (R)
(|um |q + |u|q )
"Z ≤ |h|∞
|um |
p∗
q p∗
Z
p∗
+
| u| RN
RN
q∗ # p
|Dδ (R)|
p∗ −q p∗
≤ C1 δ
p∗ −q p∗
,
(5.5)
for some constant C1 > 0. On the other hand, by the definition of Dδ (R),
Z
|h(x)| · | |um | −|u | | ≤ δ q
BCR \Dδ (R)
q
Z RN
(|um |q + |u|q ) ≤ C2 δ,
(5.6)
for some constant C2 > 0. Observe that the constants C1 and C2 are obtained via the Sobolev Embedding Theorem and the boundedness of (um ). Since um * u weakly in E, it follows from the Sobolev Embedding Theorem that, up to a subsequence, q um → u in Lloc (RN ). In particular, for R = Rδ > 0 given above, without loss of generality, we may suppose that
Z
| |um |q −|u |q | → 0,
as m → ∞.
BR
Then, taking into account the fact that h ∈ L∞ (RN ), there exists m0 ∈ N such that
Z
|h(x)| · | |um |q −|u |q | < δ,
for all m ≥ m0 .
(5.7)
BR
Thus, it follows from (5.5)–(5.7) that, for m ≥ m0 = m0 (δ),
Z RN
|h(x)| · | |um |q −|u |q | < C1 δ
p∗ −q p∗
+ (1 + C2 )δ.
Since δ > 0 can be chosen arbitrarily small, the above relation contradicts (5.4). This proves the Lemma 5.2.
Now, we may conclude the proof of Theorem A. First, we invoke Corollary 2.5 to find a sequence (um ) ⊂ E such that I ( um ) → c ≥ α > 0
and kI 0 (um )k → 0,
as m → ∞,
(5.8)
with c given by (2.1). Applying Lemma 2.7, we may assume, without loss of generality, that um * u weakly in E. From this and (2.6), it is not difficult to show that u is a critical point of I. Thus, in order to prove Theorem A, it suffices to assume that u = 0.
H.F. Lins, E.A.B. Silva / Nonlinear Analysis 71 (2009) 2890–2905
In view of Proposition 4.4, it follows that 0 < α ≤ c < p∗ Lloc
1 N
(p−N )/p N /p
|K |∞
S
2903
. Hence Proposition 3.6 can be applied to show
that um → 0 strongly in (R ). Furthermore, by Proposition 3.7, there exist a sequence (ym ) ⊂ RN and r, η > 0 such that |ym | → ∞ as m → ∞, and
Z lim sup Br (ym )
m→∞
N
|um |p ≥ η > 0,
for all m ∈ N.
(5.9)
Without loss of generality we may assume that (ym ) ⊂ ZN . Then, defining vm (y) = um (y + ym ), m ∈ N, we have kvm k0 = kum k0 for all m ∈ N. Thus, taking a subsequence if necessary, there exists v ∈ E such that vm * v weakly in p E, vm → v strongly in Lloc (RN ) and vm (x) → v(x) almost everywhere in RN . From (5.9), we have v 6= 0. We claim that v is a critical point of I0 . Indeed, given w ∈ E, in view of (V ), (K ), (g1 ), (g2 ) and (g4 ), we have I00 (vm ).w → I00 (v).w,
as m → ∞.
(5.10)
On the other hand, considering wm (y) = w(y − ym ) for m ∈ N, in view of the periodicity of V0 , K0 and g0 , we get I00 (vm ).w = I00 (um ).wm ,
for all m ∈ N.
(5.11)
Moreover, applying Lemma 5.1, we have (5.12) |I00 (um ).wm − I 0 (um ).wm | → 0, as m → ∞. By kwm k0 = kwk0 for all m ∈ N, (5.8) and (5.12), we conclude that I00 (um ).wm → 0, as m → ∞. The above relation, (5.11) and (5.10) show that v is a critical point of I0 . The claim is proved. Our next task is to verify that I0 (v) ≤ c. In order to show this fact, we apply condition (K ) and the definition of (vm ) to get 1 0 1 I (um ).um ≥ p N
I (um ) −
Z
∗
+ p K0 (x)|vm | +
RN
Z
1 p
RN
g (x, um )um − G(x, um ) .
(5.13)
Now, in view of Fatou’s lemma, taking a subsequence if necessary, we may write lim inf
1
Z
N We claim that m→∞
RN
Z lim inf m→∞
1
∗
+ p K0 (x)|vm | ≥
1 p
RN
Z
N
∗
RN
K0 (x)|v + |p .
g (x, um )um − G(x, um ) ≥
Z RN
(5.14)
1 p
g0 (x, v)v − G0 (x, v) .
(5.15)
Assuming that the claim is true, we apply (5.8), (5.13)–(5.15) and the fact that (um ) ⊂ E is bounded to get c≥
1
Z
N
∗
RN
K0 (x)|v + |p +
Z RN
1 p
g0 (x, v)v − G0 (x, v) .
Since v is a critical point of I0 , it follows from the above relation that I0 (v) ≤ c. Now, in view of (g4 )-(i), (iii), and the definition of c, c ≤ max I (tu) ≤ max I0 (tu) = I0 (u) ≤ c . t ≥0
t ≥0
This implies that there exists γ ∈ Γ such that (2.3) holds. In view of Theorem 2.3, I possesses a critical point u ∈ Kc . From c ≥ α > 0 = I (0), it follows that u is a nontrivial critical point of I. Finally, we conclude the proof of Theorem A by showing that (5.15) holds. First we observe that, in view of Lemma 5.2,
Z RN
hi (x)|um |qi → 0,
as m → ∞, i = 2, 3,
(5.16)
since hi ∈ F , p ≤ qi < p∗ and um * 0 weakly in E. Invoking (5.16) and (g4 ), we get g (x, um )um − g0 (x, um )um → 0, strongly in L1 (RN ), as m → ∞. Similarly, G(x, um ) − G0 (x, um ) → 0, strongly in L1 (RN ), as m → ∞. Consequently, by the periodicity of g0 ,
Z lim inf m→∞
RN
1 p
g (x, um )um − G(x, um ) = lim inf m→∞
Z RN
1 p
g0 (x, vm )vm − G0 (x, vm ) .
Now, observe that, by (g3 ) and (g4 ), we have 1 1 g0 (x, s)s − G0 (x, s) ≥ −h1 (x) − h2 (x)|s|q2 − h3 (x)|s|q3 . p p Invoking (5.17), Lemma 5.2 and Fatou’s Lemma, we obtain (5.15). The proof of Theorem A is complete.
(5.17)
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H.F. Lins, E.A.B. Silva / Nonlinear Analysis 71 (2009) 2890–2905
5.2. Proof of Theorem B We argue as in the initial steps of the proof of Theorem A. Since g0 satisfies (g1 ) and (g2 ), applying Corollary 2.5, we may find a sequence (um ) ⊂ E such that I 0 ( um ) → c ≥ α > 0
and kI00 (um )k → 0,
as m → ∞,
(5.18)
where c is given by (2.1). By Lemma 2.7 we may assume, without loss of generality, that um * u weakly in E. From this and (2.6), it is not difficult to show that u is a critical point of I0 . Hence, in order to prove Theorem B, it suffices to assume that (p−N )/p N /p u = 0. Since, by Proposition 4.4, 0 < α ≤ c < N1 |K |∞ S , we may apply Proposition 3.6 to conclude that um → 0 p∗
strongly in Lloc (RN ). Furthermore, by Proposition 3.7, there exist a sequence (ym ) ⊂ RN and r, η > 0 such that |ym | → ∞, as m → ∞, and
Z lim sup m→∞
Br (ym )
|um |p ≥ η > 0,
for all m ∈ N.
(5.19)
As in the proof of Theorem A, we may assume that (ym ) ⊂ ZN . Then, defining vm (y) = um (y + ym ), m ∈ N, we have kvm k0 = kum k0 for every m ∈ N. Consequently, taking a subsequence if necessary, there exists v ∈ E such that vm * v p weakly in E, vm → v strongly in Lloc (RN ) and vm (x) → v(x) almost everywhere in RN . From (5.19), we have v 6= 0. It is not difficult to show that que v is a critical point of I0 . Indeed, given w ∈ E, by (V ), (K ), (g1 ) and (g2 ), we get I00 (vm ).w → I00 (v).w,
as m → ∞.
(5.20)
On the other hand, considering wm (y) = w(y − ym ), for m ∈ N, in view of the periodicities of V0 , K0 and g0 , we get I00 (vm ).w = I00 (um ).wm ,
for all m ∈ N.
Consequently, from (5.18) and the fact that kwm k0 = kwk0 for all m ∈ N, we conclude that I00 (um ).w → 0,
as m → ∞.
This limit together with (5.20) shows that v is a critical point of I0 as claimed. The Theorem B is proved.
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