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Singularly perturbed elliptic systems
Nonlinear Analysis 64 (2006) 109 – 129 www.elsevier.com/locate/na
Singularly perturbed elliptic systems夡 Claudianor O. Alvesa , Sérgio H.M. Soaresb,∗ a Departamento de Matemática e Estatística, Universidade Federal de Campina Grande,
58109-970-Campina Grande-PB, Brazil b Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação-ICMC,
Universidade de São Paulo-USP, 13560-970, São Carlos-SP, Brazil Received 23 March 2004; accepted 10 June 2005
Abstract We study the existence and concentration behavior of positive solutions for a class of Hamiltonian systems (two coupled nonlinear stationary Schrödinger equations). Combining the Legendre–Fenchel transformation with mountain pass theorem, we prove the existence of a family of positive solutions concentrating at a point in the limit, where related functionals realize their minimum energy. In some cases, the location of the concentration point is given explicitly in terms of the potential functions of the stationary Schrödinger equations. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: 35A15; 35J50; 58J37 Keywords: Variational methods; Elliptic systems; Hamiltonian systems; Positive solutions; Concentration
1. Introduction and variational formulation Singularly perturbed elliptic equations arise in some physical and biological models. For instance, the study of standing waves solutions of the nonlinear Schrödinger equation −ih¯ 夡
j = −h¯ 2 + W (x) − ||p−1 , jt
Partially supported by Fapesp.
∗ Corresponding author. Tel.: +55 16 273 9660; fax: +55 16 273 9650.
E-mail address: [email protected] (S.H.M. Soares). 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.06.013
(1.1)
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C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
namely, solutions of the form (x, t) = exp(−iEt/h)u(x), reduces (1.1) to an elliptic ¯ equation like −2 u + V (x)u = |u|p−1 u
in RN ,
(1.2)
after rewriting the corresponding coefficients. In particular, when we consider the semiclassical limit → 0, the last equation becomes a singularly perturbed one. In addition to wide variety of models where this elliptic equation appears, the analysis of solutions of (1.2), as tends to zero, has been intensively received attention in recent years. Since the work of Floer and Weinstein [11], it is known the existence of a family of solutions u of (1.2) which develop a spike shape around one or more distinguished points of the space, while vanishing elsewhere as tends to zero. In [11], it was considered the case N = 1, p = 3, and V is assumed bounded. For a given nondegenerate critical point of V, they proved the existence of solution for (1.2) and found that this one concentrates around the critical point as → 0. In [14], Rabinowitz used a mountain-pass-type argument to find a positive ground state (least energy solution) for (1.2) for every > 0 sufficiently small, when V satisfies the global assumption lim inf V (x) > inf V (x) > 0 |x|→∞
x∈RN
and 1 < p < (N + 2)/(N − 2). In [16], Wang proved that the mountain pass solutions found in [14] concentrate around a global minimum of V as tends to 0. In [10], del Pino and Felmer found solutions which concentrate around local minima of V not necessarily nondegenerate. Motivated by the results just cited, a natural question is whether same phenomenon of concentration occurs for the singularly Hamiltonian system 2 − u + V (x)u = |v|p−1 v in RN , (1.3) −2 v + W (x)v = |u|q−1 u in RN , where N 3, V and W are assumed to be uniformly continuous and bounded positive functions defined in RN , and the numbers p > 1 and q > 1 are below the critical hyperbola, that is, 1 1 N −2 + > . p+1 q +1 N Due to coupled equations, some questions arise: Do solutions (u , v ) still concentrate? In affirmative case, the functions u and v concentrate around the same point? If so, where? The purpose of this paper is to analyze these questions. Concerning these questions, there exist some works available in the literature. In [6], Ávila and Yang established existence result for strongly indefinite semilinear elliptic system with Neumann boundary condition, and they studied the limiting behavior of the positive solutions of the singularly perturbed Hamiltonian problem. In [4], the authors in collaboration with Yang established the existence and concentration behavior for the singularly perturbed Hamiltonian system 2 − u + u = V (x)|v|p−1 v in RN , (1.4) −2 v + v = W (x)|u|q−1 u in RN .
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111
In [3], Alves and Soares studied the existence and concentration behavior for a class of the singularly perturbed gradient system. Existence problem for (1.3) was studied by de Figueiredo and Yang [9], and Sirakov [15], for more general nonlinearities. In order to derive existence results, these authors used a variational setting relies on linking type theorems for strongly indefinite functionals, which requires V ≡ W . Inspired by the method applied in [4], we used the Legendre–Fenchel transformations to reduce (1.3) into a suitable problem to apply the mountain pass theorem. It is more easier to control the critical value given by mountain pass than one from linkingtype theorems. Combining the concentration-compactness principle and the mountain pass theorem, we prove that problem (1.3) has a least energy solution (u , v ), for sufficiently small. It is showed that u and v possess just one global maximum point p , q , respectively, which for some subsequences converge to a point y ∗ , what characterizes the concentration behavior of solutions for (1.3) around at same point. The concentration point y ∗ is a global minimum of a suitable energy functional C() introduced further on. In particular, when V ≡ W , we prove that y ∗ is a global minimum of V. The present work extends the use of the method employed in [4] to Hamiltonian systems with potential functions V and W nonconstant, thus increasing the range of singularly elliptic systems where concentration behavior can be investigated. In addition, an effort is made to give a partial answer to a problem raised by Sirakov in [15] about solutions of the perturbed system (1.3) to the case of V ≡ W . Although we use the same strategy employed in [4] to derive our results, we must be more careful to introduce the variational formulation, because now the potential functions V and W are coefficients of the elliptic operators. Moreover, as the variational formulation depends of the parameter , we must do a more detailed analysis of the similar estimates found in [4]. We assume that there exist > 0 such that: V (x), W (x)
∀x ∈ RN
(H1 )
and denote by V∞ and W∞ the real numbers V∞ =
lim
|x|→+∞
V (x)
and
W∞ =
lim
|x|→+∞
W (x).
(H2 )
In order to introduce the variational formulation, we need to revise some facts involving elliptic operator. 1.1. Short review about elliptic operator Let b : RN → R be continuous and bounded function, satisfying (H1 ), and let be a positive real number. We begin recalling the invertibility of the operator − + b(x)id and the continuity of its inverse operator. In fact, let s ∈]1, ∞[. Since the operator As = − + (b(x) − 2 )id in W 2,s (RN ) has variational structure, we can prove that −As is dissipative. Then, from the characterization of the dissipative operators (see [13]), we have |(id + As )u|Ls (RN ) |u|Ls (RN )
∀ > 0,
u ∈ W 2,s (RN ).
(1.5)
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Thus, id + As is injective and has closed range, for every > 0. From [7, Theorem II. 19], this operator also is surjective for every > 0. In particular for = /2, the operator (− + b(x)id)−1 exists and | − u + b(x)u|Ls (RN ) |u|Ls (RN ) . 2
(1.6)
Thus, for every h ∈ Ls (RN ), with 1 < s < ∞, the problem −u + b(x)u = h,
x ∈ RN
(1.7)
has a unique solution u ∈ W 2,s (RN ) satisfying |h|Ls (RN ) |u|Ls (RN ) . 2
(1.8)
Define f (x) = h(x) − b(x)u(x) + u(x)
x ∈ RN .
From (1.7), the function u is solution of −u + u = f (x),
x ∈ RN .
By a priori estimate due to Agmon–Douglis–Nirenberg [1] (cf. also [5, Theorem 12.1]), there exists a positive constant c1 depending on s and N such that uW 2,s (RN ) c1 (| − u + u|Ls (RN ) + |u|Ls (RN ) ) for all u ∈ W 2,s (RN ). Then, uW 2,s (RN ) c1 (|h|Ls (RN ) + |b|L∞ (RN ) |u|Ls (RN ) + 2|u|Ls (RN ) ), which implies uW 2,s (RN ) c2 (|h|Ls (RN ) + |u|Ls (RN ) )
(1.9)
Combining (1.8) with (1.9) we get uW 2,s (RN ) c3 |h|Ls (RN ) for some positive constant c3 independent of . These facts establish the following lemma. Lemma A. Let b : RN → R be continuous and bounded function, satisfying (H1 ). Let s ∈]1, ∞[. Then for every h ∈ Ls (RN ) the problem −u + b(x)u = h,
x ∈ RN
possesses a unique solution u ∈ W 2,s (RN ). Moreover, there exits a constant K > 0 independent of such that uW 2,s (RN ) K|h|Ls (RN ) .
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
113
We introduce the linear operators R : L
p+1 p
(RN ) → W
2, p+1 p
(RN )
and S : L
q+1 q
(RN ) → W
2, q+1 q
(RN ),
where R = (− + V id)−1
S = (− + W id)−1 ,
and
with V (x) = V (x) and W (x) = W (x). By Lemma A, these operators are well defined, continuous, and their norms are uniformly bounded on . Moreover, by Sobolev imbedding theorems we can assume that R : L
p+1 p
(RN ) → Lq+1 (RN )
and S : L because
1 q+1
q+1 q
(RN ) → Lp+1 (RN )
> p1∗ =
p p+1
−
2 N
and
1 p+1
> q1∗ = 2, p+1 p
exponents of the imbedding related to W We define the linear continuous operator T : L by
q+1 q
T =
(RN ) × L
0 S
R 0
p+1 p
q q+1 N
−
2 N,
where p ∗ and q ∗ are the critical
(R ) and W
2, q+1 q
(RN ), respectively.
(RN ) → Lq+1 (RN ) × Lp+1 (RN )
that is, for w = (f, g) T w = (R g, S f ). Moreover, in what follows T w, = R g + S f, q+1 q
= (, ), w = (f, g).
p+1
(RN ) × L p (RN ) be the Banach space endowed with the norm w = |f |2q+1 + |g|2p+1 , w = (f, g) ∈ X,
Let X = L
q
hereafter |.|s and
: X → R by
h dx denote Ls -norm and RN h(x) dx, respectively. Define a functional
p
(w) = p+1
p
|g|
p+1 p
q dx + q +1
|f |
q+1 q
1 dx − 2
T w, w dx.
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C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
The functional is C 1 and its Fréchet derivative is given by 1 1 −1 −1
p |g| g dx + |f | q f dx ( ) (w) = − T w, dx, ∀ = (, ) ∈ X, here we used the relation , T w dx = w, T dx,
, w ∈ X.
We observe that if w = (f, g) is a nontrivial critical point of , then w satisfies 1
R g = |f | q
−1
and
f
1
S f = |g| p
−1
g.
Thus, if u = R g and v = S f , (u, v) is a solution of the system −u + V (x)u = |v|p−1 v in RN , −v + W (x)v = |u|q−1 u in RN .
(S )
We call throughout that w = (f, g) is the dual solution associated to (u, v). By making the change of variable x → −1 x in RN , system (S ) can be written as 2 − u + V (x)u = |v|p−1 v in RN , (S ) −2 v + W (x)v = |u|q−1 u in RN . In order to state our result, for each ∈ RN , consider the following autonomous system: −u + V ()u = |v|p−1 v, x in RN , (A ) −v + W ()v = |u|q−1 u, x in RN . The dual functional associated with (A ) is p+1 q+1 p q 1 |g| p dx + |f | q dx − T w, w dx,
(w) = p+1 q +1 2 where
T =
0 S
R 0
and R = (− + V ()id)−1
and
S = (− + W ()id)−1 .
By (H1 ) and conditions on p and q, the functional satisfies the geometric hypotheses of the mountain pass theorem, with minimax value C() given by C() = inf max ( (t)), ∈ t∈[0,1]
(1.10)
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
115
where ∈ if and only if ∈ C([0, 1], E), (0) = 0, and ( (1))0. We may verify that this number can be further characterized as C() = inf max (tu). u =0 t>0 We also denote by C∞ the minimax value related to the functional p+1 q+1 q 1 p
∞ (w) = |f | q dx − T∞ w, w dx, |g| p dx + q +1 2 p+1 where
T∞ =
0 S∞
R∞ 0
and R∞ = (− + V∞ id)−1
and
S∞ = (− + W∞ id)−1 .
By a least energy positive solution for (S ) we mean a positive strong solution (u, v) ∈
2, p+1 p
2, q+1
W (RN ) × W q (RN ) for (S ), of which the associated dual solution w = (f, g) is a critical point of the associated dual functional with least energy among all nontrivial critical points of . Our main result is as follows: Theorem 1.0.1. Suppose V and W satisfy (H1 ) and (H2 ). If C∞ > inf C(),
(C)
∈RN
then system (S ) has a least energy positive solution (u , v ) for every > 0 sufficiently small. Moreover, up to a subsequence, the respective global maximum points x and y of the functions u and v are unique and converge to the same point y ∗ , which achieves the global minimum of C(), that is, C(y ∗ ) = min C(). ∈RN
Condition (C) mentioned in Theorem 1.0.1 is not empty, see the following example. Example. If there exists ∈ RN such that V () < V∞
and
W () < W∞
then (C) occurs. In our next result, we provide the location of the concentration point. In that case, we shall need to suppose a stronger version of condition (H2 ), lim
|x|→+∞
V (x) = V∞ > inf V (x). x∈RN
2 ) implies that (C) holds. We note that in the case V ≡ W , condition (H
(Hˆ 2 )
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C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
2 ). If V ≡ W , then the respective global Corollary 1.0.2. Suppose V satisfies (H1 ) and (H maximum points x and y of the functions u and v converge, up to a subsequence, to a global minimum point of V. The proof of Theorem 1.0.1 is given in the next sections. In Section 2, we show a key proposition (see Proposition 2.0.3) that will be used later to obtain existence results. In Section 3, motivated by the arguments used in [4], we prove the existence and concentration of solutions for (S ). 2. The problem (S1 ) In order to prove the existence of least energy solutions to (S ), we first consider the existence problem for the corresponding case = 1, that is, we will study the following system
−u + V (x)u = |v|p−1 v in RN , (S1 ) −v + W (x)v = |u|q−1 u in RN . q+1
p+1
The associated functional 1 : L q (RN ) × L p (RN ) → Lq+1 (RN ) × Lp+1 (RN ) of
(S1 ) is given by p+1 q+1 q 1 p 1 p q dx + dx −
(w) = |f | T w, w dx, |g| q +1 2 p+1 w = (f, g), where
T =
0 S
R 0
and R = (− + V id)−1
and
S = (− + W id)−1 .
It is easy to verify that 1 verifies the geometry of mountain pass theorem, so there exists a sequence wn = (fn , gn ) ∈ X = L and
1 (wn ) → C
q+1 q
(RN ) × L
p+1 p
(RN ) satisfying
( 1 ) (wn ) → 0,
is the minimax level. where C Crucial step in the proof of the existence results is the following: < C ∞ hold, then (S ) has a least energy positive solution. Proposition 2.0.3. If (H1 ) and C 1 The proof of Proposition 2.0.3 will be carried out through the verification of some facts. We start proving that vanishing cannot occur to sequence wn .
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
Lemma 2.0.4. There exist r, > 0 such that either p+1 |gn | p dx , or lim sup lim sup n→∞
n→∞
Br (0)
|fn |
q+1 q
117
dx .
Br (0)
Proof. Suppose by contradiction that, for every r > 0, there is a subsequence, still denoted by wn = (fn , gn ), such that p+1 q+1 p |gn | dx = lim |fn | q dx = 0. lim n→∞
n→∞
Br (0)
Br (0)
For each n, there is tn > 0 such that
1 (tn wn ) = max 1 (tw n ). t 0
We claim that tn → 1. In fact, since ( 1 ) (tn wn )wn = 0, using that ( 1 ) (wn ) = on (1), we get ⎛ ⎞ ⎛ ⎞ p+1 q+1 1 1 ⎝ ⎠ |gn | p dx + ⎝ ⎠ |fn | q dx = on (1). − 1 − 1 (2.1) 1 1 1− p 1− q tn tn Since (wn ) is bounded in X, passing to a subsequence if necessary, we have p+1 q+1 p dx → L1 and |fn | q dx → L2 , |gn | > 0. Then, from (2.1), with either L1 = 0 or L2 = 0. If not, wn → 0, which contradicts C tn → 1. Now, since C verifies = 1 (wn ) + on (1), C we have = 1 (tn wn ) + on (1), C so that 1 (tw n ) + on (1) ∀t 0. C Thus, 2 ∞ (tw n ) + t C 2
(T∞ − T )wn , wn dx + on (1)
In particular, 2 ∞ ( n wn ) + n C 2
(T∞ − T )wn , wn dx + on (1),
∀t 0.
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C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
where n satisfies
∞ ( n wn ) = max ∞ (tw n ). t 0
Therefore, 2 C∞ + n C 2
(T∞ − T )wn , wn dx + on (1).
Now, we assert that (T∞ − T )wn , wn dx → 0
(2.2)
as n → 0.
(2.3)
Effectively, taking un = Rg n
and
u∞ n = R ∞ gn ,
we obtain, ∞ −u∞ n + V ∞ un = g n , −un + V (x)un = gn . Thus, ∞ −(u∞ n − un ) + V∞ (un − un ) = (V (x) − V∞ )un .
(2.4)
Using the fact that {wn } is a Palais–Smale sequence, we have 1
|un − (fn ) q |q+1 → 0 which implies |un |q+1 dx → 0
as n → ∞
as n → ∞.
(2.5)
Br (0)
From (2.4) and Agmon–Douglas–Nirenberg estimates, we get |u∞ n − un |W 2,q+1 C
|(V (x) − V∞ )un |q+1
1 q+1
.
Hence, by (H2 ) and (2.5), lim |u∞ n − un |q+1 = 0
n→∞
that is, lim |R∞ gn − Rg n |q+1 = 0.
n→∞
Using the Hölder’s inequality, we have lim fn (R∞ − R)gn dx = 0. n→∞
(2.6)
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
119
Similarly
lim gn (S∞ − S)fn dx = 0. n→∞
(2.7)
Therefore, by (2.6) and (2.7), we find (T∞ − T )wn , wn dx → 0 as n → 0, as desired. Now, by definition of n , we have ⎛ ⎞ ⎛ ⎞ p+1 q+1 1 1 ⎝ ⎠ |gn | p dx + ⎝ ⎠ |fn | q dx − 1 − 1 1 1 1− 1−
n p
n q =
(T∞ − T )wn , wn dx,
so that, by (2.3), ⎞ ⎛ ⎞ ⎛ p+1 q+1 1 1 p ⎠ ⎝ ⎠ ⎝ −1 |gn | dx + −1 |fn | q dx = on (1), 1 1 1− 1−
n p
n q which allows us to conclude that n → 1, as n → ∞. Taking the limit as n → ∞ in (2.2), we get C∞ , C which is a contradiction.
Proof of Proposition 2.0.3. By Lemma 2.0.4, using Sobolev imbeddings, we see that there exists a subsequence of {wn }, still denoted by {wn }, such that wn w = (f, g) in X and p+1 q+1 |g| p dx or |f | q dx . Br (0)
Br (0)
Thus, w is a nontrivial critical point of 1 . By the equation, we can conclude that g = 0 and f = 0. Then (u, v) = (Rg, Sf ) is a nontrivial solution for (S1 ). The positivity of (u, v) follows from same arguments found in [2] or [4]. In order to complete the proof of Proposition 2.0.3, it remains to prove that the solution we have w has the least energy related to the functional 1 . By the definition of C, 1 (w). C
(2.8)
Since = 1 (wn ) − 1 ( 1 ) (wn )wn + on (1), C 2 we get 1 (w). C
(2.9)
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C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
= 1 (w). So the proof of Proposition 2.0.3 is Combining (2.8) and (2.9), we find C complete. 3. The problem (S ) In the next subsections we shall prove Theorem 1.0.1. We establish the existence of least energy solutions in Section 3.1 and the limiting behavior of these solutions is studied in Section 3.2. 3.1. Existence of solution Theorem 3.1.1. Suppose (H1 ), (H2 ) and (C) hold. Then there exists 0 > 0 such that (S ) has a least energy solution for every ∈]0, 0 [. Proof. From (C), inf C() < C∞ .
∈RN
Fix z ∈ RN such that C(z) + < C ∞ , for some small > 0, and choose w0 = (f0 , g0 ) ∈ X such that
z (w0 ) = C(z) and Set
z (w0 ) = 0.
z w (x) = w0 x −
and denote
z f (x) = f0 x −
Thus,
and
z g (x) = g0 x − .
T w , w dx =
(f R g + g S f ) dx.
(3.1)
. The function u = R g satisfies −u + V (x)u = g (x) in RN . Thus, u (x) = u (x + z ) satisfies − u + V (x + z) u = g0 (x) in RN . . Similarly, v (x) = v (x + z ), where v = S f , satisfies − v + W (x + z) v = f0 (x) in RN .
(3.2)
(3.3)
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
121
Then, from (3.1)–(3.3), we have v ) dx. u + g0 T w , w dx = (f0
(3.4)
Now, set u = Rz g0 , that is, u is the solution to −u + V (z)u = g0
in RN .
(3.5)
Then, from (3.2) and Lemma A, we conclude that u u = R z g0
in W
2, p+1 p
(RN ),
as → 0.
(3.6)
(RN ),
as → 0.
(3.7)
Similarly, v v = Sz f0
in W
2, q+1 q
Then, from (3.1)–(3.7), we have T w , w dx → Tz w0 , w0 dx.
(3.8)
Let t > 0 be such that
(t w ) = max (tw ). t 0
We claim that t → 1, as → 0. Effectively, we recall that t satisfies the identity p+1 q+1 1 1 p dx + q dx = (f (g ) ) T w , w dx. 1− q1 1− p1 t t Since
(f )
we get 1 1− p1
t
q+1 q
dx =
(g0 )
p+1 p
(f0 )
dx +
q+1 q
dx
1 1− q1
and
(f0 )
q+1 q
(f )
p+1 p
dx =
(g0 )
p+1 p
(3.9)
dx
dx =
T w , w dx.
(3.10)
t
On the other hand, p+1 q+1 (g0 ) p dx + (f0 ) q dx = Tz w0 , w0 dx.
(3.11)
From (3.9)–(3.11), we find ⎛ ⎞ ⎛ ⎞ p+1 q+1 1 1 ⎝ ⎠ (g0 ) p dx + ⎝ ⎠ (f0 ) q dx − 1 − 1 1− p1 1− q1 t t =
[T w , w − Tz w0 , w0 ] dx.
(3.12)
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C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
By (3.8) and (3.12), ⎛ ⎞ ⎛ ⎞ q+1 p+1 1 1 ⎠ (f0 ) q dx = o (1), ⎝ ⎠ (g0 ) p dx + ⎝ − 1 − 1 1 1 1− 1− t q t p which allow us to conclude that t → 1, as → 0. By the definition of c , q+1 p+1 p q p+1 q+1 pt qt p c (t w ) = (f0 ) q dx dx + (g0 ) q +1 p+1 t2 − T w , w dx. 2
Hence, c z (w0 ) + o (1). Then, as → 0, we find lim sup c z (w0 ) C(z) + < C∞ . →0
Thus, there is 0 > 0 such that c < C∞
∀ ∈ (0, 0 ).
Therefore, by Proposition 2.0.3, problem (S ) has a least energy positive solution for every sufficiently small. 3.2. Concentration In this section, we study the limiting behavior of the family of solutions (u , v ), as goes to zero. We will show that this family concentrates at points where the function C() has minima. Lemma 3.2.1. There exists c > 0 such that c c
∀ ∈ (0, 0 ).
Moreover, lim sup c inf C(). →0
∈RN
Proof. We start the proof taking w = (f , g ) ∈ X such that
(w ) = c
and
( ) (w ) = 0.
By maximum principle, we have T w , w T ∗ w , w ,
(3.13)
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
where
∗
T =
0 S∗
R∗ 0
and
123
R ∗ = S ∗ = (− + id)−1 ,
with given by (H1 ). Combining the definition of c with (3.13), we get c max (tw ) max ∗ (tw ) inf max ∗ (tw) = c > 0, t 0
w∈X t 0
t 0
where c is the mountain pass level related to ∗ , where p+1 q+1 q 1 p
∗ (w) = |g| p dx + |f | q dx − T ∗ w, w dx, p+1 q +1 2 w = (f, g) ∈ X. Hence, the lower bound on c is proved. To complete the proof, fix x0 ∈ RN and let w ∈ X be the dual solution related to least energy solution (u, v) for the following system: −u + V (x0 )u = |v|p−1 v in RN , −v + W (x0 )v = |u|q−1 u in RN , that is,
x0 (w) = 0
and
x0 (w) = C(x0 ).
Let w (x) = w(x − x0 ) and t > 0 be such that c (t w ) = max (tw ). t 0
Arguing as in the proof of Theorem 3.1.1, t → 1 as → 0. So, we have
(t w ) → x0 (w)
as → 0
and hence lim sup c C(x0 ) ∀x0 ∈ RN .
(3.14)
→0
Since x0 is arbitrary, (3.14) implies lim sup c inf C(). →0
∈RN
Lemma 3.2.2. For each > 0 there exist y ∈ RN and , r > 0 such that either 2 2 lim sup (g ) p dx or lim sup (f ) q dx , →0
Br (y )
→0
Br (y )
where w = (f , g ) is the dual solution related to the solution (u , v ).
124
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
Proof. Suppose by contradiction that the lemma does not hold. Then 2 2 lim sup (f ) q dx = 0. (g ) p dx = lim sup →0
y∈RN
It implies that
y∈RN
y∈R
Br (y)
(u ) dx = lim sup 2
lim sup
→0
→0
Br (y)
N
Br (y)
→0
y∈R
N
Br (y)
(v )2 dx = 0.
By Lions’s Lemma [12], we have q+1 p+1 lim u dx = lim v dx = 0 →0
→0
which implies q+1 p+1 lim (f ) q dx = lim (g ) p dx = 0. →0
(3.15)
→0
From (3.15) c → 0
as → 0,
contrary to c c > 0.
Lemma 3.2.3. The sequence {y } is bounded. Moreover, if j yj → y ∗ , then C(y ∗ ) = inf C(). ∈RN
Proof. Suppose by contradiction that there exists a subsequence, still denoted by (y ), such that |y | → ∞. Define the functions u (x) = u (x + y )
and v (x) = v (x + y ).
Observe that these functions satisfy the following system: − u + V (x + y ) u = | v |p−1 v in RN , q−1 − v + W (x + y ) v = | u | u in RN . From Lemma A, the families {u } and {v } are bounded in W respectively, then 2, u in W u
p+1 p
(RN ) and v v in W
2, q+1 q
2, p+1 p
(RN ) and W
2, q+1 q
(RN ),
(RN ).
By Lemma 3.2.2, u 0 ( u = 0) and v 0 ( v = 0). Let w = (f , g ) be the dual solution related to ( u , v ) and w = (f, g ) is the weak limit of w in X. Since w is a solution, it follows that 1 1 ( g ) p , w dx, (3.16) g dx + (f ) q fdx = T w
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
where
0 S
R 0
with
125
= (− + V (x + y )id)−1 , R S = (− + W (x + y )id)−1 .
Then, as → 0 in (3.16), p+1 q+1 p q ( g) dx + (f ) dx = T∞ w , w dx.
(3.17)
Consequently,
∞ ( ), w ) = max ∞ (t w t 0
which implies (p − 1) C∞ ∞ ( w) = 2(p + 1)
( g)
p+1 p
(q − 1) dx + 2(q + 1)
(f)
q+1 q
dx.
By Fatou’s Lemma, C∞ lim inf →0
(p − 1) 2(p + 1)
(g )
p+1 p
dx +
(q − 1) 2(q + 1)
(f )
q+1 q
dx .
Thus, C∞ lim inf c . →0
Then Lemma 3.1 and condition C lead to a contradiction. So {y } is bounded. Employing the above arguments and supposing j yj → y ∗ , we get C(y ∗ ) lim inf cj .
(3.18)
j →∞
By Lemma 3.2.1, lim inf cj lim sup cj inf C()C(y ∗ ). N j →∞
j →∞
(3.19)
∈R
From (3.18)–(3.19), we conclude that C(y ∗ ) = inf ∈RN C().
Lemma 3.2.4. For each sufficiently small, the functions u and v decay to zero at infinity, up to a subsequence. Proof. Fix the sequence {j yj } obtained by Lemma 3.2.3 and define the functions uj (x) = uj (x + yj )
and vj (x) = vj (x + yj ).
From Lemma 3.2.2, we know that the dual solution w related to (u , v ) satisfies either 2 2 p lim sup (g ) dx , or lim sup (f ) q dx . →0
Br (y )
→0
Br (y )
126
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
Defining gj (x) = gj (x + yj ) we obtain either lim sup →0
fj (x) = fj (x + yj ),
and
2 p
( gj ) dx
or
2
(fj ) q dx .
lim sup →0
Br (0)
Br (0)
If w = (f, g ) ∈ X is the weak limit of { wj }, where w j = (fj , gj ), we get by Sobolev imbedding 2 2 q (f ) dx or ( g ) p dx . Br (0)
Br (0)
Thus, w is a nontrivial solution and f, g > 0 in RN , via maximum principle. If cj = cj , we have 1 cj = j (wj ) − ( j ) (wj )wj 2 q+1 p+1 (q − 1) (p − 1) (gj ) p dx + (fj ) q dx = 2(p + 1) 2(q + 1) p+1 q+1 (p − 1) (q − 1) p = dx + ( gj ) (fj ) q dx. 2(p + 1) 2(q + 1) By Fatou’s Lemma, (p − 1) lim inf cj j →∞ 2(p + 1)
( g)
p+1 p
(q − 1) dx + 2(q + 1)
(f)
q+1 q
(3.20)
dx = y ∗ ( w ).
(3.21)
From Lemma 3.2.1, (3.20), and (3.21), w )C(y ∗ ) = inf C(). inf C() lim sup cj lim inf cj y ∗ (
∈RN
j →∞
j →∞
∈RN
Thus, w ). lim cj = y ∗ (
j →∞
This and Fatou’s Lemma imply gj → g in L
p+1 p
(RN )
and
fj → f in L
q+1 q
(RN ).
Consequently, v in W vj →
2, q+1 q
(RN ),
2, uj → u in W
p+1 p
(RN )
(3.22)
2 sense, via bootstrap arguments (see [15]), where ( u, v) and these sequences converge in Cloc is a solution to −u + V (y ∗ )u = |v|p−1 v x ∈ RN , (Ay ∗ ) −v + W (y ∗ )v = |u|q−1 u x ∈ RN .
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
127
The functions u and v have exponential decay at infinity, as the results in [8] prove. This v decay to zero at infinity. u and and C 2 -convergence imply that j , In order to complete the proof of Theorem 1.0.1, set p qj satisfying uj ( uj (x) and pj ) = max vj ( qj ) = max vj (x). x∈RN
x∈RN
By condition (H1 ), there is > 0 such that pj ) uj (
and vj ( qj ) .
j , Since the functions uj and vj decay to zero at infinity, p qj ∈ B r (0), for some r > 0 sufficiently large. As the arguments in the proof of Lemma 3.2.4, uj → u and vj → v 2 sense, with ( u, v ) a solution of (Ay ∗ ). Thus in the Cloc
uj → u and vj → v in C 2 (Br (0)).
(3.23)
By Busca and Sirakov [8] again, ( u, v ) are radially symmetric and radially decreasing. j , Then, p qj are the unique maximum points of u, v , respectively. Recall x x u˜ j (x) = uj and v˜j (x) = vj j j form a solution of (Sj ), and that their maximum points p˜ j and q˜j are given, respectively, by j + j yj p˜ j = j p
and
q j + j y j . q˜j = j
Thus, as j yj → y ∗ , we get p˜ j , q˜j → y ∗ , with C(y ∗ ) = inf ∈RN C(). Therefore, the functions uj and vj have concentration near y ∗ . 2 ) If V ≡ W , then y ∗ is a global minimum point of Remark 3.2.5. Suppose (H1 ) and (H V. In particular, if the global minimum point of V is unique, then there exists a solution for (S ) concentrating around that point, provided is sufficiently small. 2 ), we may take x0 ∈ RN such that Proof. From (H V (x0 ) = inf V (x). x∈RN
Thus, V (x0 )V (y ∗ ). We assume that V (x0 ) < V (y ∗ ). Let (u, v) be a positive solution of the system −u + V (y ∗ )u = |v|p−1 v, x ∈ RN , −v + V (y ∗ )v = |u|q−1 u, x ∈ RN .
(Ay ∗ )
128
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
Let w = (f, g) be the dual solution related to (u, v), that is, u = Ry ∗ g
and
v = Sy ∗ f ,
where Ry ∗ = Sy ∗ = (− + V (y ∗ )id)−1 . Thus, −u + V (y ∗ )u = g, x ∈ RN , −v + V (y ∗ )v = f, x ∈ RN . Since u satisfies the equation −u + V (x0 )u = g + (V (x0 ) − V (y ∗ ))u, then Ry ∗ g = u = Rx0 g + Rx0 [(V (x0 ) − V (y ∗ ))u]. Using that (V (x0 ) − V (y ∗ ))u < 0, by the maximum principle we get Rx0 [(V (x0 ) − V (y ∗ ))u] < 0. So, (3.24)
Ry ∗ g < Rx0 g. Similarly, Sy ∗ f < Sx0 f .
(3.25)
From (3.24) and (3.25), Ty ∗ w, w < Tx0 w, w . Consequently,
y ∗ w > x0 w. Let t ∗ > 0 be such that
x0 (t ∗ w) = max x0 (tw). t 0
Thus, C(y ∗ ) = y ∗ (w) = max y ∗ (tw) y ∗ (t ∗ w) ∗
t 0
> x0 (t w) = max x0 (tw)C(x0 ) inf C() = C(y ∗ ). t 0
∈RN
which is impossible. Therefore, V (x0 ) = V (y ∗ ). Consequently, if V assumes minimum value only at one point, then y ∗ = x0 and system (1.3) has a least energy positive solution (u , v ) and both u and v concentrate at minimum of potential V.
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
129
References [1] S. Agmon, A. Douglas, L. Nirenberg, Estimates near the boundary for solutions of elliptic differential equations satisfying general boundary value conditions II, Comm. Pure Appl. Math. 12 (1959) 623–727. [2] C.O. Alves, P.C. Carrião, O.H. Miyagaki, On the existence of nontrivial solutions of a perturbed Hamiltonian systems in RN , J. Math. Anal. Appl. 276 (2002) 673–690. [3] C.O. Alves, S.H.M. Soares, On existence and concentration of solutions for a class of gradient systems, NoDEA, to appear. [4] C.O. Alves, J. Yang, S.H.M. Soares, On existence and concentration of solutions for a class of Hamiltonian systems in RN , Adv. Nonlinear Studies 3 (2003) 161–180. [5] H. Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Sc. Norm. Sup. Pisa, Ser. IV, XI (1984) 593–676. [6] A.I.Avila, J.Yang, On the existence and shape of least energy solutions for some elliptic systems, J. Differential Equations 191 (2003) 348–376. [7] H. Brezis, Analyse Fonctionelle, Masson, Paris, 1983. [8] J. Busca, B. Sirakov, Symmetry results for elliptic systems in the whole space, J. Differential Equations 163 (2000) 41–56. [9] D.G. de Figueiredo, J. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems, Nonlinear Anal. Theory Methods Appl. 3 (1998) 211–234. [10] M. Del Pino, P.L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. 4 (1996) 121–137. [11] A. Floer, A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69 (1986) 397–408. [12] P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part II, Ann. Inst. H. Poincaré Ann. Nonlineáre (1984) 223–283. [13] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, second ed., Springer, New York, 1992. [14] P.H. Rabinowitz, On a class of nonlinear Schrodinger equations, Z. Angew Math. Phys. 43 (1992) 270–291. [15] B. Sirakov, On the existence of solutions of Hamiltonian systems in RN , Adv. Differential Equations 5 (10–12) (2000) 1445–1464. [16] X. Wang, On concentration of positive solutions bounded states of nonlinear Schrodinger equations, Comm. Math. Phys. 153 (1993) 229–244.
Singularly perturbed elliptic systems夡 Claudianor O. Alvesa , Sérgio H.M. Soaresb,∗ a Departamento de Matemática e Estatística, Universidade Federal de Campina Grande,
58109-970-Campina Grande-PB, Brazil b Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação-ICMC,
Universidade de São Paulo-USP, 13560-970, São Carlos-SP, Brazil Received 23 March 2004; accepted 10 June 2005
Abstract We study the existence and concentration behavior of positive solutions for a class of Hamiltonian systems (two coupled nonlinear stationary Schrödinger equations). Combining the Legendre–Fenchel transformation with mountain pass theorem, we prove the existence of a family of positive solutions concentrating at a point in the limit, where related functionals realize their minimum energy. In some cases, the location of the concentration point is given explicitly in terms of the potential functions of the stationary Schrödinger equations. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: 35A15; 35J50; 58J37 Keywords: Variational methods; Elliptic systems; Hamiltonian systems; Positive solutions; Concentration
1. Introduction and variational formulation Singularly perturbed elliptic equations arise in some physical and biological models. For instance, the study of standing waves solutions of the nonlinear Schrödinger equation −ih¯ 夡
j = −h¯ 2 + W (x) − ||p−1 , jt
Partially supported by Fapesp.
∗ Corresponding author. Tel.: +55 16 273 9660; fax: +55 16 273 9650.
E-mail address: [email protected] (S.H.M. Soares). 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.06.013
(1.1)
110
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
namely, solutions of the form (x, t) = exp(−iEt/h)u(x), reduces (1.1) to an elliptic ¯ equation like −2 u + V (x)u = |u|p−1 u
in RN ,
(1.2)
after rewriting the corresponding coefficients. In particular, when we consider the semiclassical limit → 0, the last equation becomes a singularly perturbed one. In addition to wide variety of models where this elliptic equation appears, the analysis of solutions of (1.2), as tends to zero, has been intensively received attention in recent years. Since the work of Floer and Weinstein [11], it is known the existence of a family of solutions u of (1.2) which develop a spike shape around one or more distinguished points of the space, while vanishing elsewhere as tends to zero. In [11], it was considered the case N = 1, p = 3, and V is assumed bounded. For a given nondegenerate critical point of V, they proved the existence of solution for (1.2) and found that this one concentrates around the critical point as → 0. In [14], Rabinowitz used a mountain-pass-type argument to find a positive ground state (least energy solution) for (1.2) for every > 0 sufficiently small, when V satisfies the global assumption lim inf V (x) > inf V (x) > 0 |x|→∞
x∈RN
and 1 < p < (N + 2)/(N − 2). In [16], Wang proved that the mountain pass solutions found in [14] concentrate around a global minimum of V as tends to 0. In [10], del Pino and Felmer found solutions which concentrate around local minima of V not necessarily nondegenerate. Motivated by the results just cited, a natural question is whether same phenomenon of concentration occurs for the singularly Hamiltonian system 2 − u + V (x)u = |v|p−1 v in RN , (1.3) −2 v + W (x)v = |u|q−1 u in RN , where N 3, V and W are assumed to be uniformly continuous and bounded positive functions defined in RN , and the numbers p > 1 and q > 1 are below the critical hyperbola, that is, 1 1 N −2 + > . p+1 q +1 N Due to coupled equations, some questions arise: Do solutions (u , v ) still concentrate? In affirmative case, the functions u and v concentrate around the same point? If so, where? The purpose of this paper is to analyze these questions. Concerning these questions, there exist some works available in the literature. In [6], Ávila and Yang established existence result for strongly indefinite semilinear elliptic system with Neumann boundary condition, and they studied the limiting behavior of the positive solutions of the singularly perturbed Hamiltonian problem. In [4], the authors in collaboration with Yang established the existence and concentration behavior for the singularly perturbed Hamiltonian system 2 − u + u = V (x)|v|p−1 v in RN , (1.4) −2 v + v = W (x)|u|q−1 u in RN .
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
111
In [3], Alves and Soares studied the existence and concentration behavior for a class of the singularly perturbed gradient system. Existence problem for (1.3) was studied by de Figueiredo and Yang [9], and Sirakov [15], for more general nonlinearities. In order to derive existence results, these authors used a variational setting relies on linking type theorems for strongly indefinite functionals, which requires V ≡ W . Inspired by the method applied in [4], we used the Legendre–Fenchel transformations to reduce (1.3) into a suitable problem to apply the mountain pass theorem. It is more easier to control the critical value given by mountain pass than one from linkingtype theorems. Combining the concentration-compactness principle and the mountain pass theorem, we prove that problem (1.3) has a least energy solution (u , v ), for sufficiently small. It is showed that u and v possess just one global maximum point p , q , respectively, which for some subsequences converge to a point y ∗ , what characterizes the concentration behavior of solutions for (1.3) around at same point. The concentration point y ∗ is a global minimum of a suitable energy functional C() introduced further on. In particular, when V ≡ W , we prove that y ∗ is a global minimum of V. The present work extends the use of the method employed in [4] to Hamiltonian systems with potential functions V and W nonconstant, thus increasing the range of singularly elliptic systems where concentration behavior can be investigated. In addition, an effort is made to give a partial answer to a problem raised by Sirakov in [15] about solutions of the perturbed system (1.3) to the case of V ≡ W . Although we use the same strategy employed in [4] to derive our results, we must be more careful to introduce the variational formulation, because now the potential functions V and W are coefficients of the elliptic operators. Moreover, as the variational formulation depends of the parameter , we must do a more detailed analysis of the similar estimates found in [4]. We assume that there exist > 0 such that: V (x), W (x)
∀x ∈ RN
(H1 )
and denote by V∞ and W∞ the real numbers V∞ =
lim
|x|→+∞
V (x)
and
W∞ =
lim
|x|→+∞
W (x).
(H2 )
In order to introduce the variational formulation, we need to revise some facts involving elliptic operator. 1.1. Short review about elliptic operator Let b : RN → R be continuous and bounded function, satisfying (H1 ), and let be a positive real number. We begin recalling the invertibility of the operator − + b(x)id and the continuity of its inverse operator. In fact, let s ∈]1, ∞[. Since the operator As = − + (b(x) − 2 )id in W 2,s (RN ) has variational structure, we can prove that −As is dissipative. Then, from the characterization of the dissipative operators (see [13]), we have |(id + As )u|Ls (RN ) |u|Ls (RN )
∀ > 0,
u ∈ W 2,s (RN ).
(1.5)
112
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
Thus, id + As is injective and has closed range, for every > 0. From [7, Theorem II. 19], this operator also is surjective for every > 0. In particular for = /2, the operator (− + b(x)id)−1 exists and | − u + b(x)u|Ls (RN ) |u|Ls (RN ) . 2
(1.6)
Thus, for every h ∈ Ls (RN ), with 1 < s < ∞, the problem −u + b(x)u = h,
x ∈ RN
(1.7)
has a unique solution u ∈ W 2,s (RN ) satisfying |h|Ls (RN ) |u|Ls (RN ) . 2
(1.8)
Define f (x) = h(x) − b(x)u(x) + u(x)
x ∈ RN .
From (1.7), the function u is solution of −u + u = f (x),
x ∈ RN .
By a priori estimate due to Agmon–Douglis–Nirenberg [1] (cf. also [5, Theorem 12.1]), there exists a positive constant c1 depending on s and N such that uW 2,s (RN ) c1 (| − u + u|Ls (RN ) + |u|Ls (RN ) ) for all u ∈ W 2,s (RN ). Then, uW 2,s (RN ) c1 (|h|Ls (RN ) + |b|L∞ (RN ) |u|Ls (RN ) + 2|u|Ls (RN ) ), which implies uW 2,s (RN ) c2 (|h|Ls (RN ) + |u|Ls (RN ) )
(1.9)
Combining (1.8) with (1.9) we get uW 2,s (RN ) c3 |h|Ls (RN ) for some positive constant c3 independent of . These facts establish the following lemma. Lemma A. Let b : RN → R be continuous and bounded function, satisfying (H1 ). Let s ∈]1, ∞[. Then for every h ∈ Ls (RN ) the problem −u + b(x)u = h,
x ∈ RN
possesses a unique solution u ∈ W 2,s (RN ). Moreover, there exits a constant K > 0 independent of such that uW 2,s (RN ) K|h|Ls (RN ) .
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
113
We introduce the linear operators R : L
p+1 p
(RN ) → W
2, p+1 p
(RN )
and S : L
q+1 q
(RN ) → W
2, q+1 q
(RN ),
where R = (− + V id)−1
S = (− + W id)−1 ,
and
with V (x) = V (x) and W (x) = W (x). By Lemma A, these operators are well defined, continuous, and their norms are uniformly bounded on . Moreover, by Sobolev imbedding theorems we can assume that R : L
p+1 p
(RN ) → Lq+1 (RN )
and S : L because
1 q+1
q+1 q
(RN ) → Lp+1 (RN )
> p1∗ =
p p+1
−
2 N
and
1 p+1
> q1∗ = 2, p+1 p
exponents of the imbedding related to W We define the linear continuous operator T : L by
q+1 q
T =
(RN ) × L
0 S
R 0
p+1 p
q q+1 N
−
2 N,
where p ∗ and q ∗ are the critical
(R ) and W
2, q+1 q
(RN ), respectively.
(RN ) → Lq+1 (RN ) × Lp+1 (RN )
that is, for w = (f, g) T w = (R g, S f ). Moreover, in what follows T w, = R g + S f, q+1 q
= (, ), w = (f, g).
p+1
(RN ) × L p (RN ) be the Banach space endowed with the norm w = |f |2q+1 + |g|2p+1 , w = (f, g) ∈ X,
Let X = L
q
hereafter |.|s and
: X → R by
h dx denote Ls -norm and RN h(x) dx, respectively. Define a functional
p
(w) = p+1
p
|g|
p+1 p
q dx + q +1
|f |
q+1 q
1 dx − 2
T w, w dx.
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C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
The functional is C 1 and its Fréchet derivative is given by 1 1 −1 −1
p |g| g dx + |f | q f dx ( ) (w) = − T w, dx, ∀ = (, ) ∈ X, here we used the relation , T w dx = w, T dx,
, w ∈ X.
We observe that if w = (f, g) is a nontrivial critical point of , then w satisfies 1
R g = |f | q
−1
and
f
1
S f = |g| p
−1
g.
Thus, if u = R g and v = S f , (u, v) is a solution of the system −u + V (x)u = |v|p−1 v in RN , −v + W (x)v = |u|q−1 u in RN .
(S )
We call throughout that w = (f, g) is the dual solution associated to (u, v). By making the change of variable x → −1 x in RN , system (S ) can be written as 2 − u + V (x)u = |v|p−1 v in RN , (S ) −2 v + W (x)v = |u|q−1 u in RN . In order to state our result, for each ∈ RN , consider the following autonomous system: −u + V ()u = |v|p−1 v, x in RN , (A ) −v + W ()v = |u|q−1 u, x in RN . The dual functional associated with (A ) is p+1 q+1 p q 1 |g| p dx + |f | q dx − T w, w dx,
(w) = p+1 q +1 2 where
T =
0 S
R 0
and R = (− + V ()id)−1
and
S = (− + W ()id)−1 .
By (H1 ) and conditions on p and q, the functional satisfies the geometric hypotheses of the mountain pass theorem, with minimax value C() given by C() = inf max ( (t)), ∈ t∈[0,1]
(1.10)
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
115
where ∈ if and only if ∈ C([0, 1], E), (0) = 0, and ( (1))0. We may verify that this number can be further characterized as C() = inf max (tu). u =0 t>0 We also denote by C∞ the minimax value related to the functional p+1 q+1 q 1 p
∞ (w) = |f | q dx − T∞ w, w dx, |g| p dx + q +1 2 p+1 where
T∞ =
0 S∞
R∞ 0
and R∞ = (− + V∞ id)−1
and
S∞ = (− + W∞ id)−1 .
By a least energy positive solution for (S ) we mean a positive strong solution (u, v) ∈
2, p+1 p
2, q+1
W (RN ) × W q (RN ) for (S ), of which the associated dual solution w = (f, g) is a critical point of the associated dual functional with least energy among all nontrivial critical points of . Our main result is as follows: Theorem 1.0.1. Suppose V and W satisfy (H1 ) and (H2 ). If C∞ > inf C(),
(C)
∈RN
then system (S ) has a least energy positive solution (u , v ) for every > 0 sufficiently small. Moreover, up to a subsequence, the respective global maximum points x and y of the functions u and v are unique and converge to the same point y ∗ , which achieves the global minimum of C(), that is, C(y ∗ ) = min C(). ∈RN
Condition (C) mentioned in Theorem 1.0.1 is not empty, see the following example. Example. If there exists ∈ RN such that V () < V∞
and
W () < W∞
then (C) occurs. In our next result, we provide the location of the concentration point. In that case, we shall need to suppose a stronger version of condition (H2 ), lim
|x|→+∞
V (x) = V∞ > inf V (x). x∈RN
2 ) implies that (C) holds. We note that in the case V ≡ W , condition (H
(Hˆ 2 )
116
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
2 ). If V ≡ W , then the respective global Corollary 1.0.2. Suppose V satisfies (H1 ) and (H maximum points x and y of the functions u and v converge, up to a subsequence, to a global minimum point of V. The proof of Theorem 1.0.1 is given in the next sections. In Section 2, we show a key proposition (see Proposition 2.0.3) that will be used later to obtain existence results. In Section 3, motivated by the arguments used in [4], we prove the existence and concentration of solutions for (S ). 2. The problem (S1 ) In order to prove the existence of least energy solutions to (S ), we first consider the existence problem for the corresponding case = 1, that is, we will study the following system
−u + V (x)u = |v|p−1 v in RN , (S1 ) −v + W (x)v = |u|q−1 u in RN . q+1
p+1
The associated functional 1 : L q (RN ) × L p (RN ) → Lq+1 (RN ) × Lp+1 (RN ) of
(S1 ) is given by p+1 q+1 q 1 p 1 p q dx + dx −
(w) = |f | T w, w dx, |g| q +1 2 p+1 w = (f, g), where
T =
0 S
R 0
and R = (− + V id)−1
and
S = (− + W id)−1 .
It is easy to verify that 1 verifies the geometry of mountain pass theorem, so there exists a sequence wn = (fn , gn ) ∈ X = L and
1 (wn ) → C
q+1 q
(RN ) × L
p+1 p
(RN ) satisfying
( 1 ) (wn ) → 0,
is the minimax level. where C Crucial step in the proof of the existence results is the following: < C ∞ hold, then (S ) has a least energy positive solution. Proposition 2.0.3. If (H1 ) and C 1 The proof of Proposition 2.0.3 will be carried out through the verification of some facts. We start proving that vanishing cannot occur to sequence wn .
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
Lemma 2.0.4. There exist r, > 0 such that either p+1 |gn | p dx , or lim sup lim sup n→∞
n→∞
Br (0)
|fn |
q+1 q
117
dx .
Br (0)
Proof. Suppose by contradiction that, for every r > 0, there is a subsequence, still denoted by wn = (fn , gn ), such that p+1 q+1 p |gn | dx = lim |fn | q dx = 0. lim n→∞
n→∞
Br (0)
Br (0)
For each n, there is tn > 0 such that
1 (tn wn ) = max 1 (tw n ). t 0
We claim that tn → 1. In fact, since ( 1 ) (tn wn )wn = 0, using that ( 1 ) (wn ) = on (1), we get ⎛ ⎞ ⎛ ⎞ p+1 q+1 1 1 ⎝ ⎠ |gn | p dx + ⎝ ⎠ |fn | q dx = on (1). − 1 − 1 (2.1) 1 1 1− p 1− q tn tn Since (wn ) is bounded in X, passing to a subsequence if necessary, we have p+1 q+1 p dx → L1 and |fn | q dx → L2 , |gn | > 0. Then, from (2.1), with either L1 = 0 or L2 = 0. If not, wn → 0, which contradicts C tn → 1. Now, since C verifies = 1 (wn ) + on (1), C we have = 1 (tn wn ) + on (1), C so that 1 (tw n ) + on (1) ∀t 0. C Thus, 2 ∞ (tw n ) + t C 2
(T∞ − T )wn , wn dx + on (1)
In particular, 2 ∞ ( n wn ) + n C 2
(T∞ − T )wn , wn dx + on (1),
∀t 0.
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C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
where n satisfies
∞ ( n wn ) = max ∞ (tw n ). t 0
Therefore, 2 C∞ + n C 2
(T∞ − T )wn , wn dx + on (1).
Now, we assert that (T∞ − T )wn , wn dx → 0
(2.2)
as n → 0.
(2.3)
Effectively, taking un = Rg n
and
u∞ n = R ∞ gn ,
we obtain, ∞ −u∞ n + V ∞ un = g n , −un + V (x)un = gn . Thus, ∞ −(u∞ n − un ) + V∞ (un − un ) = (V (x) − V∞ )un .
(2.4)
Using the fact that {wn } is a Palais–Smale sequence, we have 1
|un − (fn ) q |q+1 → 0 which implies |un |q+1 dx → 0
as n → ∞
as n → ∞.
(2.5)
Br (0)
From (2.4) and Agmon–Douglas–Nirenberg estimates, we get |u∞ n − un |W 2,q+1 C
|(V (x) − V∞ )un |q+1
1 q+1
.
Hence, by (H2 ) and (2.5), lim |u∞ n − un |q+1 = 0
n→∞
that is, lim |R∞ gn − Rg n |q+1 = 0.
n→∞
Using the Hölder’s inequality, we have lim fn (R∞ − R)gn dx = 0. n→∞
(2.6)
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
119
Similarly
lim gn (S∞ − S)fn dx = 0. n→∞
(2.7)
Therefore, by (2.6) and (2.7), we find (T∞ − T )wn , wn dx → 0 as n → 0, as desired. Now, by definition of n , we have ⎛ ⎞ ⎛ ⎞ p+1 q+1 1 1 ⎝ ⎠ |gn | p dx + ⎝ ⎠ |fn | q dx − 1 − 1 1 1 1− 1−
n p
n q =
(T∞ − T )wn , wn dx,
so that, by (2.3), ⎞ ⎛ ⎞ ⎛ p+1 q+1 1 1 p ⎠ ⎝ ⎠ ⎝ −1 |gn | dx + −1 |fn | q dx = on (1), 1 1 1− 1−
n p
n q which allows us to conclude that n → 1, as n → ∞. Taking the limit as n → ∞ in (2.2), we get C∞ , C which is a contradiction.
Proof of Proposition 2.0.3. By Lemma 2.0.4, using Sobolev imbeddings, we see that there exists a subsequence of {wn }, still denoted by {wn }, such that wn w = (f, g) in X and p+1 q+1 |g| p dx or |f | q dx . Br (0)
Br (0)
Thus, w is a nontrivial critical point of 1 . By the equation, we can conclude that g = 0 and f = 0. Then (u, v) = (Rg, Sf ) is a nontrivial solution for (S1 ). The positivity of (u, v) follows from same arguments found in [2] or [4]. In order to complete the proof of Proposition 2.0.3, it remains to prove that the solution we have w has the least energy related to the functional 1 . By the definition of C, 1 (w). C
(2.8)
Since = 1 (wn ) − 1 ( 1 ) (wn )wn + on (1), C 2 we get 1 (w). C
(2.9)
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C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
= 1 (w). So the proof of Proposition 2.0.3 is Combining (2.8) and (2.9), we find C complete. 3. The problem (S ) In the next subsections we shall prove Theorem 1.0.1. We establish the existence of least energy solutions in Section 3.1 and the limiting behavior of these solutions is studied in Section 3.2. 3.1. Existence of solution Theorem 3.1.1. Suppose (H1 ), (H2 ) and (C) hold. Then there exists 0 > 0 such that (S ) has a least energy solution for every ∈]0, 0 [. Proof. From (C), inf C() < C∞ .
∈RN
Fix z ∈ RN such that C(z) + < C ∞ , for some small > 0, and choose w0 = (f0 , g0 ) ∈ X such that
z (w0 ) = C(z) and Set
z (w0 ) = 0.
z w (x) = w0 x −
and denote
z f (x) = f0 x −
Thus,
and
z g (x) = g0 x − .
T w , w dx =
(f R g + g S f ) dx.
(3.1)
. The function u = R g satisfies −u + V (x)u = g (x) in RN . Thus, u (x) = u (x + z ) satisfies − u + V (x + z) u = g0 (x) in RN . . Similarly, v (x) = v (x + z ), where v = S f , satisfies − v + W (x + z) v = f0 (x) in RN .
(3.2)
(3.3)
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
121
Then, from (3.1)–(3.3), we have v ) dx. u + g0 T w , w dx = (f0
(3.4)
Now, set u = Rz g0 , that is, u is the solution to −u + V (z)u = g0
in RN .
(3.5)
Then, from (3.2) and Lemma A, we conclude that u u = R z g0
in W
2, p+1 p
(RN ),
as → 0.
(3.6)
(RN ),
as → 0.
(3.7)
Similarly, v v = Sz f0
in W
2, q+1 q
Then, from (3.1)–(3.7), we have T w , w dx → Tz w0 , w0 dx.
(3.8)
Let t > 0 be such that
(t w ) = max (tw ). t 0
We claim that t → 1, as → 0. Effectively, we recall that t satisfies the identity p+1 q+1 1 1 p dx + q dx = (f (g ) ) T w , w dx. 1− q1 1− p1 t t Since
(f )
we get 1 1− p1
t
q+1 q
dx =
(g0 )
p+1 p
(f0 )
dx +
q+1 q
dx
1 1− q1
and
(f0 )
q+1 q
(f )
p+1 p
dx =
(g0 )
p+1 p
(3.9)
dx
dx =
T w , w dx.
(3.10)
t
On the other hand, p+1 q+1 (g0 ) p dx + (f0 ) q dx = Tz w0 , w0 dx.
(3.11)
From (3.9)–(3.11), we find ⎛ ⎞ ⎛ ⎞ p+1 q+1 1 1 ⎝ ⎠ (g0 ) p dx + ⎝ ⎠ (f0 ) q dx − 1 − 1 1− p1 1− q1 t t =
[T w , w − Tz w0 , w0 ] dx.
(3.12)
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C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
By (3.8) and (3.12), ⎛ ⎞ ⎛ ⎞ q+1 p+1 1 1 ⎠ (f0 ) q dx = o (1), ⎝ ⎠ (g0 ) p dx + ⎝ − 1 − 1 1 1 1− 1− t q t p which allow us to conclude that t → 1, as → 0. By the definition of c , q+1 p+1 p q p+1 q+1 pt qt p c (t w ) = (f0 ) q dx dx + (g0 ) q +1 p+1 t2 − T w , w dx. 2
Hence, c z (w0 ) + o (1). Then, as → 0, we find lim sup c z (w0 ) C(z) + < C∞ . →0
Thus, there is 0 > 0 such that c < C∞
∀ ∈ (0, 0 ).
Therefore, by Proposition 2.0.3, problem (S ) has a least energy positive solution for every sufficiently small. 3.2. Concentration In this section, we study the limiting behavior of the family of solutions (u , v ), as goes to zero. We will show that this family concentrates at points where the function C() has minima. Lemma 3.2.1. There exists c > 0 such that c c
∀ ∈ (0, 0 ).
Moreover, lim sup c inf C(). →0
∈RN
Proof. We start the proof taking w = (f , g ) ∈ X such that
(w ) = c
and
( ) (w ) = 0.
By maximum principle, we have T w , w T ∗ w , w ,
(3.13)
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
where
∗
T =
0 S∗
R∗ 0
and
123
R ∗ = S ∗ = (− + id)−1 ,
with given by (H1 ). Combining the definition of c with (3.13), we get c max (tw ) max ∗ (tw ) inf max ∗ (tw) = c > 0, t 0
w∈X t 0
t 0
where c is the mountain pass level related to ∗ , where p+1 q+1 q 1 p
∗ (w) = |g| p dx + |f | q dx − T ∗ w, w dx, p+1 q +1 2 w = (f, g) ∈ X. Hence, the lower bound on c is proved. To complete the proof, fix x0 ∈ RN and let w ∈ X be the dual solution related to least energy solution (u, v) for the following system: −u + V (x0 )u = |v|p−1 v in RN , −v + W (x0 )v = |u|q−1 u in RN , that is,
x0 (w) = 0
and
x0 (w) = C(x0 ).
Let w (x) = w(x − x0 ) and t > 0 be such that c (t w ) = max (tw ). t 0
Arguing as in the proof of Theorem 3.1.1, t → 1 as → 0. So, we have
(t w ) → x0 (w)
as → 0
and hence lim sup c C(x0 ) ∀x0 ∈ RN .
(3.14)
→0
Since x0 is arbitrary, (3.14) implies lim sup c inf C(). →0
∈RN
Lemma 3.2.2. For each > 0 there exist y ∈ RN and , r > 0 such that either 2 2 lim sup (g ) p dx or lim sup (f ) q dx , →0
Br (y )
→0
Br (y )
where w = (f , g ) is the dual solution related to the solution (u , v ).
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C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
Proof. Suppose by contradiction that the lemma does not hold. Then 2 2 lim sup (f ) q dx = 0. (g ) p dx = lim sup →0
y∈RN
It implies that
y∈RN
y∈R
Br (y)
(u ) dx = lim sup 2
lim sup
→0
→0
Br (y)
N
Br (y)
→0
y∈R
N
Br (y)
(v )2 dx = 0.
By Lions’s Lemma [12], we have q+1 p+1 lim u dx = lim v dx = 0 →0
→0
which implies q+1 p+1 lim (f ) q dx = lim (g ) p dx = 0. →0
(3.15)
→0
From (3.15) c → 0
as → 0,
contrary to c c > 0.
Lemma 3.2.3. The sequence {y } is bounded. Moreover, if j yj → y ∗ , then C(y ∗ ) = inf C(). ∈RN
Proof. Suppose by contradiction that there exists a subsequence, still denoted by (y ), such that |y | → ∞. Define the functions u (x) = u (x + y )
and v (x) = v (x + y ).
Observe that these functions satisfy the following system: − u + V (x + y ) u = | v |p−1 v in RN , q−1 − v + W (x + y ) v = | u | u in RN . From Lemma A, the families {u } and {v } are bounded in W respectively, then 2, u in W u
p+1 p
(RN ) and v v in W
2, q+1 q
2, p+1 p
(RN ) and W
2, q+1 q
(RN ),
(RN ).
By Lemma 3.2.2, u 0 ( u = 0) and v 0 ( v = 0). Let w = (f , g ) be the dual solution related to ( u , v ) and w = (f, g ) is the weak limit of w in X. Since w is a solution, it follows that 1 1 ( g ) p , w dx, (3.16) g dx + (f ) q fdx = T w
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
where
0 S
R 0
with
125
= (− + V (x + y )id)−1 , R S = (− + W (x + y )id)−1 .
Then, as → 0 in (3.16), p+1 q+1 p q ( g) dx + (f ) dx = T∞ w , w dx.
(3.17)
Consequently,
∞ ( ), w ) = max ∞ (t w t 0
which implies (p − 1) C∞ ∞ ( w) = 2(p + 1)
( g)
p+1 p
(q − 1) dx + 2(q + 1)
(f)
q+1 q
dx.
By Fatou’s Lemma, C∞ lim inf →0
(p − 1) 2(p + 1)
(g )
p+1 p
dx +
(q − 1) 2(q + 1)
(f )
q+1 q
dx .
Thus, C∞ lim inf c . →0
Then Lemma 3.1 and condition C lead to a contradiction. So {y } is bounded. Employing the above arguments and supposing j yj → y ∗ , we get C(y ∗ ) lim inf cj .
(3.18)
j →∞
By Lemma 3.2.1, lim inf cj lim sup cj inf C()C(y ∗ ). N j →∞
j →∞
(3.19)
∈R
From (3.18)–(3.19), we conclude that C(y ∗ ) = inf ∈RN C().
Lemma 3.2.4. For each sufficiently small, the functions u and v decay to zero at infinity, up to a subsequence. Proof. Fix the sequence {j yj } obtained by Lemma 3.2.3 and define the functions uj (x) = uj (x + yj )
and vj (x) = vj (x + yj ).
From Lemma 3.2.2, we know that the dual solution w related to (u , v ) satisfies either 2 2 p lim sup (g ) dx , or lim sup (f ) q dx . →0
Br (y )
→0
Br (y )
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C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
Defining gj (x) = gj (x + yj ) we obtain either lim sup →0
fj (x) = fj (x + yj ),
and
2 p
( gj ) dx
or
2
(fj ) q dx .
lim sup →0
Br (0)
Br (0)
If w = (f, g ) ∈ X is the weak limit of { wj }, where w j = (fj , gj ), we get by Sobolev imbedding 2 2 q (f ) dx or ( g ) p dx . Br (0)
Br (0)
Thus, w is a nontrivial solution and f, g > 0 in RN , via maximum principle. If cj = cj , we have 1 cj = j (wj ) − ( j ) (wj )wj 2 q+1 p+1 (q − 1) (p − 1) (gj ) p dx + (fj ) q dx = 2(p + 1) 2(q + 1) p+1 q+1 (p − 1) (q − 1) p = dx + ( gj ) (fj ) q dx. 2(p + 1) 2(q + 1) By Fatou’s Lemma, (p − 1) lim inf cj j →∞ 2(p + 1)
( g)
p+1 p
(q − 1) dx + 2(q + 1)
(f)
q+1 q
(3.20)
dx = y ∗ ( w ).
(3.21)
From Lemma 3.2.1, (3.20), and (3.21), w )C(y ∗ ) = inf C(). inf C() lim sup cj lim inf cj y ∗ (
∈RN
j →∞
j →∞
∈RN
Thus, w ). lim cj = y ∗ (
j →∞
This and Fatou’s Lemma imply gj → g in L
p+1 p
(RN )
and
fj → f in L
q+1 q
(RN ).
Consequently, v in W vj →
2, q+1 q
(RN ),
2, uj → u in W
p+1 p
(RN )
(3.22)
2 sense, via bootstrap arguments (see [15]), where ( u, v) and these sequences converge in Cloc is a solution to −u + V (y ∗ )u = |v|p−1 v x ∈ RN , (Ay ∗ ) −v + W (y ∗ )v = |u|q−1 u x ∈ RN .
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
127
The functions u and v have exponential decay at infinity, as the results in [8] prove. This v decay to zero at infinity. u and and C 2 -convergence imply that j , In order to complete the proof of Theorem 1.0.1, set p qj satisfying uj ( uj (x) and pj ) = max vj ( qj ) = max vj (x). x∈RN
x∈RN
By condition (H1 ), there is > 0 such that pj ) uj (
and vj ( qj ) .
j , Since the functions uj and vj decay to zero at infinity, p qj ∈ B r (0), for some r > 0 sufficiently large. As the arguments in the proof of Lemma 3.2.4, uj → u and vj → v 2 sense, with ( u, v ) a solution of (Ay ∗ ). Thus in the Cloc
uj → u and vj → v in C 2 (Br (0)).
(3.23)
By Busca and Sirakov [8] again, ( u, v ) are radially symmetric and radially decreasing. j , Then, p qj are the unique maximum points of u, v , respectively. Recall x x u˜ j (x) = uj and v˜j (x) = vj j j form a solution of (Sj ), and that their maximum points p˜ j and q˜j are given, respectively, by j + j yj p˜ j = j p
and
q j + j y j . q˜j = j
Thus, as j yj → y ∗ , we get p˜ j , q˜j → y ∗ , with C(y ∗ ) = inf ∈RN C(). Therefore, the functions uj and vj have concentration near y ∗ . 2 ) If V ≡ W , then y ∗ is a global minimum point of Remark 3.2.5. Suppose (H1 ) and (H V. In particular, if the global minimum point of V is unique, then there exists a solution for (S ) concentrating around that point, provided is sufficiently small. 2 ), we may take x0 ∈ RN such that Proof. From (H V (x0 ) = inf V (x). x∈RN
Thus, V (x0 )V (y ∗ ). We assume that V (x0 ) < V (y ∗ ). Let (u, v) be a positive solution of the system −u + V (y ∗ )u = |v|p−1 v, x ∈ RN , −v + V (y ∗ )v = |u|q−1 u, x ∈ RN .
(Ay ∗ )
128
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
Let w = (f, g) be the dual solution related to (u, v), that is, u = Ry ∗ g
and
v = Sy ∗ f ,
where Ry ∗ = Sy ∗ = (− + V (y ∗ )id)−1 . Thus, −u + V (y ∗ )u = g, x ∈ RN , −v + V (y ∗ )v = f, x ∈ RN . Since u satisfies the equation −u + V (x0 )u = g + (V (x0 ) − V (y ∗ ))u, then Ry ∗ g = u = Rx0 g + Rx0 [(V (x0 ) − V (y ∗ ))u]. Using that (V (x0 ) − V (y ∗ ))u < 0, by the maximum principle we get Rx0 [(V (x0 ) − V (y ∗ ))u] < 0. So, (3.24)
Ry ∗ g < Rx0 g. Similarly, Sy ∗ f < Sx0 f .
(3.25)
From (3.24) and (3.25), Ty ∗ w, w < Tx0 w, w . Consequently,
y ∗ w > x0 w. Let t ∗ > 0 be such that
x0 (t ∗ w) = max x0 (tw). t 0
Thus, C(y ∗ ) = y ∗ (w) = max y ∗ (tw) y ∗ (t ∗ w) ∗
t 0
> x0 (t w) = max x0 (tw)C(x0 ) inf C() = C(y ∗ ). t 0
∈RN
which is impossible. Therefore, V (x0 ) = V (y ∗ ). Consequently, if V assumes minimum value only at one point, then y ∗ = x0 and system (1.3) has a least energy positive solution (u , v ) and both u and v concentrate at minimum of potential V.
C.O. Alves, S.H.M. Soares / Nonlinear Analysis 64 (2006) 109 – 129
129
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