Existence and multiplicity of solutions for a class of elliptic systems in Rn

Nonlinear Analysis 71 (2009) 2585–2599

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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na

Existence and multiplicity of solutions for a class of elliptic systems in Rn Paulo Rabelo Departamento de Matemática, Universidade Federal de Sergipe, 49100-000, São Cristóvão, SE, Brazil

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Article history: Received 31 March 2008 Accepted 15 January 2009

Using minimax methods we study the existence and multiplicity of nontrivial solutions for semilinear elliptic systems of the form

MSC: 35J20 35J60 35Q55

The potentials ai (x) for i = 1, . . . , m may change sign, control the growth of nonlinearities and can be ‘‘large’’ at the infinity. We treat both superquadratic situation and nonquadratic situation at infinity on the nonlinearities fi (x, u1 , . . . , um ) for i = 1, . . . , m. © 2009 Elsevier Ltd. All rights reserved.

−∆ui + ai (x)ui = fi (x, u1 , . . . , um ) for x ∈ Rn and i = 1, . . . , m.

Keywords: Elliptic systems Schrödinger equation Variational methods Solitary waves

1. Introduction In this paper we consider a class of semilinear elliptic systems of the form

−∆ui + ai (x)ui = fi (x, u1 , . . . , um ) for x ∈ Rn and i = 1, . . . , m, n

(P) n

m

where the functions ai : R → R are continuous being able to assume negatives values and the nonlinearities fi : R × R → R are also continuous with fi (x, 0, . . . , 0) = 0, can be unbounded in x ∈ Rn and have growth controlled by the growth of the potentials for i = 1, . . . , m. We shall consider the variational situation in which f = (f1 , . . . , fm ) = ∇ F for some function F : Rn × Rm → R of class C 1 , where ∇ F stands for the gradient of F in the variables U = (u1 , . . . , um ) ∈ Rm . On Rm we use the usual euclidian scalar product h·, ·i with the associate norm k · k = h·, ·i1/2 . We will write the system above in the form

−∆U + A(x)U = f (x, U ), where ∆ = diag(∆, . . . , ∆) and A(x) = diag(a1 (x), . . . , am (x)). The problem (P) appears in many areas of mathematical physics, in particular, it is well know that it arises naturally in connection with solitons (standing wave) solutions of nonlinear Schrodinger equations

∂φ = −∆φ + V (x)φ − g (x, |φ|)φ, x ∈ Rn , t > 0, ∂t where φ is a vector function, φ = (φ1 , . . . , φm ), satisfying the system of equations above and we are interested in solutions in the form φj (x, t ) = exp(−iωj t )uj (x), for some ωj ∈ R and uj real functions. Replacing into above system and setting aj (x) = V (x) − ωj and fj = g (x, |φ|)uj leads to a system of real elliptic partial differential equations like (P) which has a formal variational structure whose amplitude ui (z ) vanishes at infinity for i = 1, . . . , m. i

E-mail address: [email protected]. 0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.101

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Systems of the nonlinear Schrödinger type have been widely used in the applied sciences but a mathematical study of standing wave solutions was undertaken only very recently, prompted in particular by the discovery of the importance of these systems as models in nonlinear optics (see for example [1,2]) and in the study of Bose–Einstein condensates (see [3]). Motivated by the results of Rabinowitz [4] and Costa [5] that showed existence of a nontrivial solution under the assumption that the potentials are positives and coercive, and by the results of Sirakov [6] that, in the scalar case, showed existence of a nontrivial solution considering potentials assuming negatives values, we extend the existence result from Costa [5] for a elliptic system where potentials assume negative values and nonlinearities have unbounded growth. We suppose that the potentials ai are continuous and satisfy the following assumptions.

(A1 ) There exist D > 0 such that ai (x) ≥ −D, for all x ∈ Rn and i = 1, . . . , m. Next, in order to apply variational setting, we consider the following subspace of H 1 (Rn , Rm )

 E=

U ∈ H 1 (Rn , Rm ) :



Z Rn

A(x)U · U dx < ∞ ,

which is a Hilbert space when endowed with the scalar product

hU , V iE =

Z Rn

[∇ U · ∇ V + A(x)U · V ] dx 1/2

to which corresponds the norm kU kE = hU , U iE . Here, as usual, H 1 (Rn , Rm ) denotes the Sobolev spaces modeled in L2 (Rn , Rm ) with norm

kU k2H 1 (Rn ,Rm ) =

m Z X

Rn

i=1

(|∇ ui |2 + |ui |2 ) dx.

To ensure the imbedding of E into H 1 (Rn , Rm ) assume the following condition on the first eigenvalue of the operator −∆ + A(x):

(A2 ) λ1 = infU ∈E , kU k2 =1

R

Rn

[|∇ U |2 + A(x)U · U ] dx > 0.

We use the following notation. If Ω ⊂ Rn is open and 2 ≤ s < 2n/(n − 2), we set

νs (Ω ) =

Z inf

U ∈H 1 (Ω ,Rm ) 0 kU ks =1

Ω

  |∇ U |2 + A(x)U · U dx,

and we put νs (∅) = +∞. The hypothesis below provide the compact imbedding of E into some weighted Lebesgue space.

(A3 ) limR→+∞ νs (Rn \ BR ) = +∞. The function K (x) that will control the growth of the nonlinear term f is given by the following hypothesis n (A4 ) There exist a function K (x) ∈ L∞ loc (R ), with K (x) ≥ 1, and constants α > 1, c0 , R0 > 0 such that "  1/α # + K (x) ≤ c0 1 + min ai (x) for all |x| ≥ R0 , 1≤i≤m

n where a+ i (x) = maxx∈R {0, ai (x)} for all i = 1, . . . , m.

We introduce the following assumptions on the nonlinear term:

(f1 ) The function f satisfies the growth condition |f (x, U )| ≤ cK (x)(1 + |U |p ) for all (x, U ) ∈ Rn × Rm , where c > 0, 1 ≤ p < p# ≤ (n + 2)/(n − 2) if n ≥ 3 or 1 ≤ p < ∞ if n = 1, 2 (we will determine later what it means p# ); (f2 ) |f (x, U )|/K (x) = o(|U |) as U → 0 uniformly in x ∈ Rn . Let us consider firstly the superquadratic case, that is,

(f3 ) There is a constant µ > 2 such that 0 < µF (x, U ) ≤ U · f (x, U ) for all (x, U ) ∈ Rn × (Rm \ {0}). Using assumption (f3 ) we prove that the Palais–Smale condition is indeed satisfied and together with the growth hypothesis we obtain the geometric shape required by the standard mountain-pass theorem (see [7,8]). This is our first main result: Theorem 1.1. Suppose (A1 )–(A4 ) and (f1 )–(f3 ) are satisfied, with s = p + 1 in (A3 ). Then (P) has a strong solution U ∈ C 1 (Rn , Rm ) ∩ W 1,2 (Rn , Rm ) that decay at infinity. If, in addition, f (x, U ) is odd in U, then (P) has infinitely many solutions.

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Our next result considers the situation in which the spectrum of the operator −∆ + A(x) is crossed as |U | varies from 0 to ∞ and we replace the Ambrosetti-Rabinowitz condition by a hypothesis of nonquadraticity at infinity introduced by CostaMagalhães in [9] which is a sufficient condition to obtain the Cerami compactness condition. More precisely, we will assume that

(f4 ) There are θ > 0 and a > 0 such that U · f (x, U ) − 2F (x, U ) ≥ a|U |θ > 0

for all (x, U ) ∈ Rn × (Rm \ {0}).

In this case, we state the second main result on the existence of a nonzero solution for the problem (P). Theorem 1.2. Under the hypotheses of Theorem 1.1, with (f3 ) replaced by (f4 ) and 2θ > α n(p − 1)/(α − 1) if n ≥ 2 or θ > α(p − 1)/(α − 1) if n = 1, assume, in addition, that F satisfies the crossing conditions

(f5 ) lim sup|U |→0

2F (x,U ) |U |2

≤ a < λk < b ≤ lim inf|U |→+∞

2F (x,U ) |U |2

uniformly in x ∈ Rn ;

(f6 ) F (x, U ) ≥ 21 λk−1 |U |2 for all (x, U ) ∈ Rn × Rm . Then the same conclusions of the previous theorem hold. Remark 1.3. (1) Assumptions like (A1 )–(A4 ) were already used in [6] in order to study the scalar problem, which correspond to m = 1. (2) Following the same idea in [6], we can check that a sufficient condition for the hypothesis (A3 ) is that

! m \ i AM \ (BR ) = 0 for all M > 0, lim R→∞ i =1 where AiM = {x ∈ Rn : ai (x) < M }. Thus, the potentials satisfying V (x) ≥ 1 and 1/V (x) ∈ L1 (Rn ) or such that, for every M > 0, the set {x ∈ Rn : V (x) < M } has a finite measure of Lebesgue, also satisfy the conditions (A1 ) and (A3 ). The potential V (x) = x21 x22 . . . x2n − A, with any constant A > 0 chosen so that λ1 > 0, satisfies the conditions (A1 ) and (A3 ) and does not satisfy the assumptions above. (3) Notice that the Theorem 1.1 is a special case of the Theorem 1.2 in the autonomous case, that is, when F (x, U ) = F (U ) F (U ) and lim|U |→0 |U |µ > 0. (4) See [5] for an example of a nonlinearity f (x, u) satisfying assumption (f4 ) but not (f3 ) for the scalar problem, that is, m = 1. (5) The hypothesis (f1 ) shows a relationship of dependence between the potential A(x) and nonlinearity f (x, U ) so that the growth of f (x, U ) also imposes restrictions on the potentials. For example, the function f (x, u1 , . . . , um ) = pω(x)(|u1 |p−1 u1 , . . . , |um |p−1 um ), with ω(x) ≥ β > 0, satisfies all the conditions of Theorem 1.2 provided that ai (x) ≥ [ω(x)]α , for |x| > R0 and i = 1, . . . , m. (6) It is readily seen that using classical regularity arguments for elliptic equations one can prove the regularity and decay of solutions of (P) (see [10]). The paper is organized as follows. In Section 2 we introduce a variational setting and we obtain some compactness properties of the functional associated to the system (P). We reserve the Section 3 for the proof of the main results. Notation.— In this paper we make use of the following notation:

• C , C0 , C1 , C2 , . . . denote positive (possibly different) constants. • BR denotes the open ball centered at origin and radius R > 0. • For 1 ≤ p < ∞, Lp (Rn , Rm ) denotes the Lebesgue spaces with norm kU k2p =

m X

kui k2Lp (Rn ) .

i=1

• • • •

C0∞ (Rn ) denotes the space of infinitely differentiable functions with compact support. X ∗ is the topological dual of the Banach space X . By h·, ·i we denote the duality pairing between X ∗ and X . If Ω ⊂ Rn is a measurable set, then |Ω | denotes his Lebesgue measure in Rn .

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2. Preliminary results Our choice of the variational setting E ensures that the embedding in H 1 (Rn , Rm ) is continuous and that the functional Φ : E → R given by 1

Φ (U ) =

2 1

=

2

kU k2E −

Z

F (x, U ) dx

Rn

kU k2E − N (U ),

is well-defined and of class C 1 . This is the content of the next two lemmas. Lemma 2.1. Suppose that (A1 ) and (A2 ) are satisfied. Then E is a Hilbert space continuously embedded into H 1 (Rn , Rm ). Proof. We claim that there exist a constant ζ > 0 such that

kU k ≥ ζ 2 E

Z Rn

|∇ U |2 dx,

for all U ∈ E. Indeed, assume by contradiction that the claim does not hold. Then there exist a sequence {Uk } ⊂ E such that

kUk k2E ≤

1 k

Z Rn

|∇ Uk |2 dx.

1 Letting Wk = k∇ Uk k− 2 Uk , we obtain

Z Rn

|∇ Wk |2 dx = 1 and kWk k2E ≤

1 k

.

By (A2 ) we get

λ1 kWk k22 ≤ kWk k2E ≤

1 k

.

Since λ1 > 0, it follows that kWk k2 → 0. On the other hand, using (A1 ), we obtain

Z

|Wk |2 dx ≤

−D

Z

Rn

Rn

A(x)Wk · Wk dx

= kWk k2E − ≤

1 k

Z

|∇ Wk |2 dx Rn

− 1.

This implies that kWk k22 ≥ 1/2D > 0, for k ∈ N large, which is a contradiction. Thus,

Z |∇ U |2 dx + λ1 |U |2 dx Rn Rn Z ≥ min{ζ , λ1 } (|∇ U |2 + |U |2 ) dx

2kU k2E ≥ ζ

Z

Rn

shows that the embedding of E in H 1 (RN , Rm ) is continuous. Now we prove that E is complete. We suppose that {Uk } is a Cauchy sequence in E. By continuity of the embedding of E in H 1 (RN , Rm ) we have that {Uk } is a Cauchy sequence in H 1 (Rn , Rm ), and so there exist U ∈ H 1 (Rn , Rm ) such that

kUk − U kH 1 (Rn ,Rm ) → 0. Thus, there exists a subsequence {Ukj } and h ∈ L2 (Rn ) such that Ukj (x) → U (x) and |Ukj (x)| ≤ h(x) almost everywhere in Rn , for all j ∈ N. Since

Z Rn

A(x)U · U dx ≤

Z Rn

A+ (x)U · U dx,

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we may assume that ai (x) > 0 for all x ∈ Rn and i = 1, . . . , m. Notice that 1/2

kA

Z

(Uki − Ukj )k = 2 2

A(x)(Uki − Ukj ) · (Uki − Ukj ) dx

Rn

≤ kUki − Ukj k2E , implies that {A1/2 Ukj } is a Cauchy sequence in L2 (Rn , Rm ) and so we can to extract a subsequence such that, for all r ≥ 1 integers,

kA1/2 Ur +1 − A1/2 Ur k2 ≤

1 2r

.

Now, letting gk (x) =

k X 1/2 A (x) (Ur +1 (x) − Ur (x)) , r =1

we have by Minkowski’s inequality that

kg k k2 ≤

k X

kA1/2 (Ur +1 − Ur )k2 ≤ 1.

r =1

Therefore, using the Monotone Convergence Theorem we conclude that gk (x) converge almost everywhere in Rn for a finite limit g (x) ∈ L2 (Rn ). Since for each ` ∈ N we have

|A1/2 (x)(Ur +` (x) − Ur (x))| ≤ gr +`−1 (x) − gr −1 (x) almost everywhere in Rn , thus taking the limit as ` → ∞, we obtain that

|A1/2 (x)(U (x) − Ur (x))| ≤ g (x) − gr −1 (x) ≤ g (x) almost everywhere in Rn . Hence

|A1/2 (x)U (x)| ≤ g (x) − |A1/2 (x)Ur (x)| almost everywhere in Rn , and consequently A1/2 U ∈ L2 (Rn , Rm ). This implies that U ∈ E. It remains to prove that Uk → U in E. This follows by the convergence of {Uk } in H 1 (Rn , Rm )

Z Rn

|∇(Uk − U )|2 dx → 0

and the fact that

kA1/2 (Uk − U )k22 =

Z Rn

A(x)(Uk − U ) · (Uk − U ) dx → 0. 

Lemma 2.2. Assume that (A1 )–(A2 ), (A4 ) and (f1 )–(f2 ) are satisfied. Then the functional Φ is well defined and class C 1 on E. Furthermore, for all ε > 0 there exist Cε > 0 such that

Z Rn

|F (x, U )| dx ≤ εkU k2E + Cε kU kpE+1 .

(2.1)

Proof. By (f2 ), given ε > 0 there exists δ > 0 such that |fi (x, U )| ≤ ε K (x)|U | always that |U | < δ , for each i = 1, . . . , m fixed. Now, for |U | ≥ δ , it follows by (f1 ) that

|fi (x, U )| ≤ c0 K (x)(1 + |U |p )   1 + 1 = c0 K (x)|U |p |U |p   1 ≤ c0 K (x) p + 1 |U |p . δ Thus,

|fi (x, U )| ≤ K (x)(ε|U | + Cε |U |p ) n

m

uniformly in x ∈ R , for all U ∈ R .

(2.2)

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Let ξ (t ) = F (x, tU ) with t ∈ [0, 1]. Then, by the mean value theorem, there exists a number θ ∈ (0, 1) such that |ξ (1) − ξ (0)| = |ξ 0 (θ )|, that is, m X |F (x, U )| = Di F (x, tu1 , . . . , θ ui , . . . , tum )ui i=1 m X

≤

|fi (x, tu1 , . . . , θ ui , . . . , tum )||ui |,

i =1

which in combination with (2) implies that

|F (x, U )| ≤ K (x)(ε|U | + Cε |U |p )|U | = K (x)(ε|U |2 + Cε |U |p+1 ).

(2.3)

Now, by (A4 ) we get

Z

K (x)|U | dx = s

Rn

Z

K (x)|U | dx + s

Z

"

Z ≤

K (x)|U |s dx |x|≤R0

|x|>R0

c0 1 + |x|>R0

" s s

≤ C kU k +

1/α # Z min ai (x) |U |s dx + max {K (x)}



+

1≤i≤m

|U |s dx |x|≤R0

#

m Z X i=1

|x|≤R0

1/α

ai (x) +

|ui | dx . s

|x|>R0

Using Hölder’s inequality, we obtain

Z

1/α |ui |s dx ≤ a+ i (x)

Z

2 a+ i (x)|ui | dx

1/α Z

|x|>R0

|x|>R0

|ui |(αs−2)/(α−1) dx

(α−1)/α

,

(2.4)

|x|>R0

and by (A1 ) we have

Z

ai (x)|ui | dx = +

2

Z Rn

|x|>R0

Z ≤

ai (x)|ui | dx − 2

Z

ai (x)|ui | dx − 2

|x|≤R0

Z

2 a− i (x)|ui | dx

|x|>R0

ai (x)u2i + Du2i dx.



Rn



(2.5)

From (2.4), (2.5) and (A3 ) we obtain

Z

h

K (x)|U |s dx ≤ C kU kss + kU k2E + DkU k22

Rn

"



s s

≤ C kU k + 1 +

D


1/α

1/α kU k

λ1

(α s−2)/α

kU k(αs−2)/(α−1)

2/α E

i

(α s−2)/α kU k(αs−2)/(α−1)

# .

(2.6)

Thus, the space E can be embedded into the space LsK (x)



(R , R ) := U : R → R mensurable : n

m

n

m

Z

K (x)|U | dx < +∞ s

Rn



provided that (α s − 2)/(α − 1) < 2∗ . In particular, for s = p + 1, we have that p < p# =

n+2 n−2

−

4

α(n − 2)

.

(2.7)

We conclude that

Z Rn

K (x)|U |s dx ≤ c kU ksE ,

for all 2 ≤ s < p# + 1, and

Z Rn

|F (x, U )| dx ≤ ε

Z

K (x)|U | dx + Cε 2

Rn

Z Rn

K (x)|U |p+1 dx

≤ εkU k2E + Cε kU kpE+1 . This expression show that the functional Φ is well defined.

P. Rabelo / Nonlinear Analysis 71 (2009) 2585–2599

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Our next goal is to show that Φ is of class C 1 on E. We note that the first term of Φ is C 1 with Gáteaux derivative hU , V iE . Now define γ : [0, 1] → R by γ (z ) = F (x, U + tzV ), where V = (v1 , . . . , vm ). Then, by the mean value theorem, there exist θ ∈ (0, 1) such that γ (1) − γ (0) = γ 0 (θ ). That is, F (x, U + tV ) − F (x, U ) =

m X

Di F (x, U + θ tV )t vi

i=1

=

m X

fi (x, U + θ tV )t vi .

i=1

By (2.2) we have that 1 t

(F (x, U + tW ) − F (x, U )) ≤ K (x)|V |(|U | + |V |) + K (x)C |V |(|U |p + |V |p ) ≤ CK (x)[|U |2 + |U |p+1 + |V |2 + |V |p+1 ].

Since the term on the right-hand side is integrable, we can apply the Dominated Convergence Theorem to conclude that 1

hN 0 (U ), V i = lim [N (U + tV ) − N (U )] t →0 t Z X m = fi (x, U )vi dx Rn i=1

Z = Rn

f (x, U ) · V dx.

As N 0 (U ) is linear and bounded, it suffices to show that the Gáteaux derivative of N is continuous. Let Uk → U in E. Then Uk → U in Ls (BR , Rm ), for all 2 ≤ s ≤ 2∗ and R > 0. Consequently, Uk (x) → U (x) almost everywhere in Rn and there exist h(x) ∈ Ls (Rn ) such that |Uk (x)| ≤ h(x) almost everywhere in Rn . Given W ∈ E, we define Gk (x) = W (x) · f (x, Uk (x)). Then, by (2),

|Gk (x)| ≤ |W ||f (x, Uk )| ≤ K (x)(|Uk | + C1 |Uk |p )|W |  
|W |2 |h(x)|2 ≤ K (x) + + C1 |W |p+1 + |h(x)|p+1 2

2

= m(x), with m(x) ∈ L1 (Rn ). By the Dominated Convergence Theorem, Gk (x) → G(x) = W · f (x, U ) in L1 (Rn ) and so

Z lim

k→+∞

Rn

W · f (x, Uk ) dx =

Z Rn

W · f (x, U ) dx.

Hence, for each W ∈ E with kW kE = 1, we get

kN 0 (Uk ) − N 0 (U )kE ∗ = sup |hN 0 (Uk ) − N 0 (U ), W i| kW kE =1 Z = sup W · [f (x, Uk ) − f (x, U )] dx → 0, kW kE =1 Rn

and the proof of Lemma 2.2 is complete.



Remark 2.3. We notice that from (7) we can see that p# → 2∗ − 1 as α → +∞. Thus, our result extends the main theorem in Costa [5], where the potentials are coercive and we can take α → +∞ and K (x) a positive constant. We observe that critical points of Φ are weak solutions of the system because 0 = hΦ 0 (U ), V i = hU , V iE −

Z Rn

f (x, U ) · V dx,

for all V ∈ E. In particular, for (0, . . . , 0, vi , 0, . . . , 0) ∈ E we have that, for all i = 1, . . . , m,

Z Rn

[∇ ui ∇vi + ai (x)ui vi − fi (x, U )vi ] dx = 0.

The compactness of the embedding E into LsK (x) (Rn , Rm ) is established by our next theorem.

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P. Rabelo / Nonlinear Analysis 71 (2009) 2585–2599

Proposition 2.4. Suppose that (A1 )–(A4 ) and (f1 )–(f2 ) hold. Then the embedding of E into LsK (x) (Rn , Rm ) is compact, for all 2 ≤ s < p# + 1.

Proof. The continuous embedding was established in the proof of Lemma 2.1. Let us show that (A3 ) is sufficient for the embedding to be compact. We suppose Uk * 0 in E. Considering the embedding E ,→ H 1 (Rn , Rm ) ,→ H 1 (BR , Rm ) ,→ Ls (BR , Rm ), we have that Uk → 0 in Ls (BR , Rm ), for all 2 ≤ s < 2∗ and R > 0. Let φ ∈ C ∞ (Rn ) such that 0 ≤ φ ≤ 1, φ ≡ 0 on BR and φ ≡ 1 on Rn \ BR+1 . Then


kUk kss = C k(1 − φ)Uk kss + kφ Uk kss "Z Z (1 − φ)s |Uk |s dx + =C BR+1

# φ |Uk | dx . s

Rn \BR

s

The first term tends to zero when k → +∞ and we denote it by βk . Now, letting 1 Wk = kφ Uk k− s φ Uk ,

we have that Wk ∈ H01 (Rn \ BR , Rm ) and kWk ks = 1. By the definition of νs (Ω ) it follows that

νs (R \ BR )kφ n

Z

Uk 2s

k ≤ Rn \BR

|∇(φ Uk )|2 + A(x)(φ Uk ) · (φ Uk ) dx

and, in consequence,

kUk kss ≤ βk +

1

νs (

Rn

\ BR )s/2

kφ Uk ksE = βk + γR ,

where γR → 0 as R → +∞ by (A3 ). Thus, Uk → 0 in Ls (Rn , Rm ), for all 2 ≤ s < 2∗ . By the expression (2.6) in the Lemma 2.2 we have that

" kU k

s LsK (x) (Rn ,Rm )

s s



≤ C kU k + 1 +

D

1/α

λ1

kU k

2/α E

(α s−2)/α kU k(αs−2)/(α−1)

#

for any U ∈ E. Then we infer that Uk → 0 in LsK (x) (Rn , Rm ) with 2 ≤ s < 2∗ − 4(α(n − 2))−1 .



The next two propositions shows that Φ satisfies a compactness condition of the Palais–Smale type. Proposition 2.5. Assume that (A1 )–(A4 ) and (f1 )–(f3 ) hold. Then, with s = p + 1 in (A3 ), the functional Φ satisfy the Palais–Smale condition on E. Proof. We first prove that if Uk * U in E, then

Z Rn

[f (x, Uk ) − f (x, U )] · (Uk − U ) dx → 0

as k → +∞. p+1

Indeed, it follows of the Lemma 2.2 that Uk → U in LK (x) (Rn , Rm ) and by the inverse Lebesgue theorem, we can find a p+1 subsequence, still denoted by {Uk }, and a function h ∈ LK (x) (Rn ) such that

|Uk (x)| ≤ h(x) and Uk (x) → U (x) almost everywhere in Rn . Since {Uk } is bounded in L2K (x) (Rn , Rm ), setting Hk (x) = |f (x, Uk (x)) − f (x, U (x))||Uk (x) − U (x)| we have that Hk (x) → 0 almost everywhere in Rn and by Young’s inequality and (2.3), we obtain that

|Hk (x)| ≤ ε[K (x)(|Uk |2 + |U |2 )] + Cε [K (x)(|Uk |p+1 + |U |p+1 ) + |Uk |p |U | + |U |p |Uk |] ≤ C1 [K (x)(|Uk |2 + |U |2 )] + C2 K (x)|h(x)|p+1 = C1 ωk + C2 g , where ωk , g ∈ L1 (Rn ) and kωk kL1 (Rn ) ≤ M. As g is integrable, for each δ > 0 there exist r1 > 0 such that

Z

g (x) dx < |x|>r1

δ 2C2

.

P. Rabelo / Nonlinear Analysis 71 (2009) 2585–2599

2593

Similarly, for each k ∈ N, there exist Rk > 0 such that

Z

ωk (x) dx < |x|>Rk

δ 2C1

.

Since Uk → U in L2K (x) (Rn , Rm ), there exist k0 ∈ N such that, for all k > k0 ,

kUk kL2

K (x)

(Rn ,Rm )

≤ kU kL2

K (x)

(Rn ,Rm )

+

δ 4C1

.

Thus, taking r2 > 0 such that

Z

δ

K (x)|U |2 dx <

8C1

|x|>r2

,

it follows that, for all k > k0 ,

Z

ωk (x) dx ≤

Z

|x|>r2

K (x)|Uk |2 dx +

Z

|x|>r2

Z

K (x)|U |2 dx +

≤2 |x|>r2

≤

δ 2C1

K (x)|U |2 dx |x|>r2

δ 4C1

.

Choosing R = max{r1 , r2 , R1 , . . . , Rk0 } for all k ∈ N we obtain

Z

Hk (x) dx = C1 |x|>R

Z

ωk (x) dx + C2 |x|>R

Z

g (x) dx < δ. |x|>R

We next verify that for all δ > 0 we can find r > 0 such that, for any S ⊂ Rn with |S | < r, we have

kHk kL1 (S ) < δ for all k ∈ N. That is, {Hk } is uniformly integrable. Indeed, letting

 δ r = min , , 2MC1 2C2 kg kL1 (Rn ) 

δ

it follows that

Z

Hk (x) dx ≤ C1 S

Z

ωk (x) dx + C2 S

Z

g (x) dx S

≤ C1 M |S | + C2 |S |kg kL1 (Rn ) < δ. Therefore, we can apply the Vitali convergence theorem to conclude that Hk → 0 in L1 (Rn ). Now if {Uk } ⊂ E is such that |Φ (Uk )| ≤ K and kΦ 0 (Uk )kE ∗ → 0, then



1 2

−

1

µ



kUk k2E ≤ Φ (Uk ) −

1

µ

Φ 0 (Uk )Uk ≤ K + C kUk kE

which implies that {Uk }E is bounded in E and has a subsequence weakly convergent. Since 1 2

kUk − U k2E = hΦ 0 (Uk ) − Φ 0 (U ), Uk − U i +

Z Rn

we conclude that {Uk } has a subsequence convergent.

[f (x, Uk ) − f (x, U )] · (Uk − U ) dx,

(2.8)



Next, we recall the Cerami compactness condition introduced in Definition 2.6. A functional Φ ∈ C 1 (E , R) is said to satisfy the Cerami compactness condition if any sequence {Uk } ⊂ E such that Φ (Uk ) → c and (1 + kUk kE )kΦ 0 (Uk )kE ∗ → 0, possesses a convergent subsequence. Proposition 2.7. Under the hypotheses of Proposition 2.5, with (f3 ) replaced by (f4 ) and 2θ > α n(p − 1)/(α − 1) if n ≥ 2 or θ > α(p − 1)/(α − 1) if n = 1, Φ satisfy the Cerami compactness condition. Proof. We give the proof only for the case n ≥ 3, the case n = 1, 2 being similar. Let {Uk } ⊂ E such that |Φ (Uk )| ≤ K and (1 + kUk kE )kΦ 0 (Uk )kE ∗ → 0. We claim that {Uk } has a bounded subsequence in E. To obtain a contradiction suppose that

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P. Rabelo / Nonlinear Analysis 71 (2009) 2585–2599

kUk kE → +∞. Using (f4 ) we obtain that Z [Uk · f (x, Uk ) − 2F (x, Uk )] dx ≥ akU kθθ 2Φ (Uk ) − Φ 0 (Uk )Uk = Rn

and, on the other hand, 2Φ (Uk ) − Φ 0 (Uk )Uk ≤ 2|Φ (Uk )| + kΦ 0 (Uk )kE ∗ kUk kE ≤ K1 . Hence, for all k ∈ N,

kUk kθ ≤ K2 .

(2.9)

Writing Qk (x) = Uk (x) · f (x, Uk (x)) − 2F (x, Uk (x)) we have that

Z lim sup Rn

k→+∞

Qk (x) dx ≤ K1 .

(2.10)

Now, by (2.3) we find that 1 2

kUk k2E − Φ (Uk ) =

Z Rn

≤ε

F (x, Uk (x)) dx

Z

K (x)|Uk | dx + C1 2

Rn

Z Rn

K (x)|Uk |p+1 dx,

and substituting (2.6) we see that 1 2

"



k k − Φ (Uk ) ≤ ε k k + 1 + Uk 2E

Uk 22

D

λ1

 α1

# Uk 2E

k k

" p+1 Uk p+1

+ C2 k k

 + 1+

D

 α1

λ

2

Uk Eα

k k kUk k

α(p+1)−2 α α(p+1)−2 α−1

α(p+1)−2

There is no loss of generality in assuming that θ < min{p + 1, α−1 } < 2∗ (the case θ > max{p + 1, follows another without restriction on θ ). Thus, by the interpolation inequality,

# .

α(p+1)−2 } α−1

(2.11)

> 2 it

kU kp+1 ≤ kU k1θ −t kU kt2∗ and kU k α(p+1)−2 ≤ kU kθ1−s kU ks2∗ α−1

θ

2∗

for all U ∈ L (R , R ) ∩ L (R , R ), with 1 p+1

n

=

m

1−t

θ

+

t 2∗

n

m

and

α−1 1−s s = + ∗. α(p + 1) − 2 θ 2

(2.12)

∗

Using the continuous embedding E ,→ L2 (Rn , Rm ) we obtain



1 2



−ε 1+

D

λ1



(1−s) α(p+α1)−2

kUk k2E − Φ (Uk ) ≤ εkUk k22 + C2 kUk k(θ1−t )(p+1) kUk ktE(p+1) + C3 kUk kθ

2

kUk kEα

+s α(p+α1)−2

,

and from there, for ε > 0 sufficiently small, 2

kUk k2E ≤ K1 + K2 kUk k22 + K3 kUk ktE(p+1) + K4 kUk kEα

+s α(p+α1)−2

(2.13) α(p+1)−2 s α

According to relation (2.12) we obtain that t (p + 1) < 2 and α2 + < 2 since 2θ > nα(p − 1)/(α − 1). e ∈ E such Letting Wk = Uk /kUk kE and using the compact embedding of E in L2 (Rn , Rm ) we conclude that there exist W e in E and Wk → W e in L2 (Rn , Rm ). Thus, Wk (x) → W e (x) almost everywhere in Rn . Now, that, up to subsequence, Wk * W dividing (2.13) by kUk k2E and passing to the limit, we obtain

e k22 . 1 ≤ K4 k W e | 6= 0 and implies that the set S = {x ∈ Rn : |W e (x)| 6= 0} has a positive measure. Since Qk (x) ≥ a|Uk (x)|θ > 0 This gives |W and |Uk (x)| → +∞ for x ∈ S, it follows of Fatou’s Lemma that Z lim inf k→+∞

Rn

Qk (x) dx ≥ lim inf k→+∞

Z

Qk (x) dx S

Z ≥ a lim inf |Uk (x)|θ dx k→+∞ S Z ≥ a lim inf |Uk (x)|θ dx → +∞. S k→+∞

This contradicts (2.10). Therefore, using (2.8) we obtain a subsequence convergent.



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The compact embedding of E in L2K (x) (Rn , Rm ) implies the following result. Lemma 2.8. The spectrum of the operator ∆ + A(x) on E consist of a sequence {λk } of eigenvalues such that λk → +∞ as k → +∞. Proof. For each U ∈ E we define the linear functional S : E → R by S (W ) = hU , W iL2 (Rn ,Rm ) . Then by Riesz representation theorem, there exist T (U ) ∈ E such that

hT (U ), W iE = S (W ) = hU , W iL2 (Rn ,Rm ) . Thus, the operator T : E → E is linear, bounded, symmetric and positive definite. By the compact embedding of E in L2 (Rn , Rm ) it follows that T is compact. Writing the problem of eigenvalues

∆U + A(x)U = λU as

hU , W iE = λhT (U ), W iE for all W ∈ E , we have that T (U ) = λ−1 U and from there λn → +∞ as k → +∞.



The next proposition is technical and will be used in the proof of the Theorem 1.2. Proposition 2.9. Assume (A1 )–(A4 ), (f1 )–(f2 ) and (f5 ) hold. Then for all b b ∈ (λk , b) we have N (U ) − b bkU k22

lim inf

kU k2E

kU kE →+∞

≥ 0.

Proof. By (f5 ) there exist R > 0 such that F (x, U ) ≥ b b|U |2 for all x ∈ Rn and |U | > R. Taking ΩR = {x ∈ Rn : |U (x)| < R}, we have that

Z

N (U ) =

ΩR

Z ≥ ΩR

Z

F (x, U ) dx +

Rn \ΩR

F (x, U ) dx + b b



=

Z

ΩR

F (x, U ) dx

Z

|U |2 dx Rn \ΩR

F (x, U ) − b b|U |2 dx + b bkU k22 .



Hence, it is enough to show that NR (U )

≥ 0, kU k2E R where NR (U ) = Ω [F (x, U ) − b b|U |2 ] dx. We claim that limkU kE →+∞ R lim inf

kU kE →+∞

NR (U )

kU k2E

= 0. Indeed, by contradiction, suppose that

there exists δ0 > 0 and a sequence {Uk } in E such that kUk kE → +∞ and |NR (Uk )| ≥ δ0 kUk k2E for all k ∈ N. We assume that NR (Uk ) ≥ 0 (the case NR (Uk ) < 0 being similar). Let Wk = Uk /kUk kE . Since kWk kE = 1 and the embedding of E in e ∈ E such that L2K (x) (Rn , Rm ) is compact, there exist W

e in E Wk * W e in L2K (x) (Rn , Rm ) Wk → W e (x) almost everywhere in Rn Wk (x) → W |Wk (x)| ≤ h(x) ∈ L2K (x) (Rn ). Letting Qk (x) =



F (x, Uk (x))

|Uk (x)|2

 −b b χk (x)|Wk (x)|2 ,

where χk is the characteristic function of the set

 Ω (R, k) = x ∈ Rn : 0 < |Uk (x)| < R ,

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P. Rabelo / Nonlinear Analysis 71 (2009) 2585–2599

we have that

Z Rn

Qk (x)dx =



Z Ω (R,k)

F (x, Uk (x))

|Uk (x)|2

 |Uk |2 b −b dx ≥ δ0 > 0 kUk k2E

(2.14)

for all k ∈ N. On the other hand, as h ∈ L2 (Rn ) and

  F (x, Uk (x)) 2 b |Qk (x)| ≤ − b h ( x ) 2 |Uk (x)|
≤ ε K (x) + Cε K (x)|Uk (x)|p−1 − b b h2 (x)
≤ ε K (x) + Rp−1 Cε K (x) − b b h2 (x) , e (x)| = 0} we have it follows that Qk ∈ L1 (Rn ). Moreover, Qk → 0 almost everywhere in Rn since on the set {x ∈ Rn : |W e (x)| > 0, then |Uk (x)| = kUk kE |Wk (x)| → +∞. Thus, χk (x) = 0 for large k. Therefore, by |Wk (x)| → 0, whereas, if |W R Lebesgue’s theorem, we conclude that Rn Qk (x) dx → 0, which contradicts (2.14).  3. Proofs of Theorems 1.1 and 1.2 Now we are in a position of giving a proof of the theorems announced in the introduction. The proof is divided in several steps. Existence of critical points for Φ under assumptions of Theorem 1.1. We suppose that |U | ≥ 1. We shall make use of the spherical polar coordinate representation U = (ρ, φ) = (ρ, φ1 , . . . , φm−1 ), where ρ ≥ 1, −π ≤ φ1 ≤ π , 0 ≤ φ2 , . . . , φm−1 ≤ π and u1 = ρ sin(φ1 ) sin(φ2 ) · · · sin(φm−1 ), u2 = ρ cos(φ1 ) sin(φ2 ) · · · sin(φm−1 ), u3 = ρ cos(φ2 ) · · · sin(φm−1 ),

.. . um = ρ cos(φm−1 ). Substituting in hypothesis (f3 ), we get µF (x, U ) ≤ ρ Fρ (x, U ) and thence F (x, U ) ≥





min F (x, V ) |U |µ > 0

|V |=1

(3.15)

for all x ∈ Rn and |U | ≥ 1. Hence, given any bounded set S ⊂ Rn , there exist C = C (S ) > 0 such that F (x, U ) ≥ C |U |µ

(3.16)

for all x ∈ S and |U | ≥ 1, whence

Φ (U ) ≤

1 2

µ

kU k2E − C kU kLµ (S ) .

This shows that there exist many e ∈ E such that Φ (e) < 0. Now, using the embedding of E in Ls (Rn , Rm ) for 2 ≤ s < p# + 1 we have that   1 Φ (U ) ≥ − ε kU k2E − Cε kU kpE+1 2 and taking ε = 1/4 and choosing r > 0 so that 1/4 − Cε r p−1 > 1/8, we get

Φ (U ) ≥

1

kU k2E 8 for all kU kE ≤ r. Therefore, the mountain-pass geometry holds and considering that Φ is of class C 1 and satisfy the Palais–Smale condition, we can use the Mountain Pass Theorem to conclude the existence of a critical point U ∈ E of the functional Φ with Φ (U ) > 0 (see [7,8]). In the other words, the problem (P) has a nonzero weak solution, and the proof of existence of Theorem 1.1 is complete. Existence of critical points for Φ under assumptions of Theorem 1.2. Our proof starts with a suitable decomposition of the 1 space E. Let Nk−1 = {φ1k−1 , . . . , φjkk− } orthonormal base of the eigenspace corresponding to eigenvalue λk−1 of the operator −1

−∆ + A(x) and we denote by Eλ+k−1 , Eλ0k−1 and Eλ−k−1 the subspaces of E where I − λk−1 T is positive definite, zero and negative

P. Rabelo / Nonlinear Analysis 71 (2009) 2585–2599

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definite, respectively. The operator T being definite in the Lemma 2.8. Thus, E = (Eλ−k−1 ⊕ Eλ0k−1 ) ⊕ Eλ+k−1 = E − ⊕ E + . Notice that if i ≤ k − 1 we have 0 = kφji k2E − λi kφji k22 ≥ kφji k2E − λk−1 kφji k22 , whence φji ∈ E − and if i > k − 1 we have that φji ∈ E + . Thus, dim(E − ) < +∞. Now, choosing b a > 0 and b b > 0 such that 2F (x, U )

lim sup

|U |2

|U |→0

≤ a
2F (x, U )

|U |→∞

|U |2

,

(3.17)

we have that there exist b δ > 0 such that F (x, U ) ≤ (b a/2)|U |2 whenever |U | < b δ . If |U | ≥ b δ , then proceeding as in the proof of the expression (3), we verify that F (x, U ) ≤ K (x)Cε |U |p+1 . Hence,

b a

F (x, U ) ≤

2

|U |2 + K (x)Cε |U |p+1 p+1

for all x ∈ Rn and U ∈ Rm . Using the embedding of E in LK (x) (Rn , Rm ) we get that

Φ (U ) ≥

1 2

b kU kp+1 for all U ∈ E . (kU k2E −b akU k22 ) − M E

akU k22 ≤ 0 then U ⊥φji for all Notice that Eba+ = Eλ+k−1 . Indeed, Eba+ ⊂ Eλ+k−1 and if there is U ∈ Eλ+k−1 \ Eba+ such that kU k2E − b i ≤ k − 1. By the characterization of the eigenvalue λk it follows that

λk ≤

kU k2E ≤b a, kU k22

b > 0 such that which contradicts (3.17). Thus, we can take m bkU k2E kU k2E −b akU k22 ≥ m for all U ∈ E + , for otherwise there exists a sequence {Uk } in E + such that

kUk k2E −b akUk k22 ≤

1 n

kUk k2E

and from there λk ≤ b a. But this contradicts the hypothesis that b a < λk . Therefore,

b kU kp−1 )kU k2E Φ (U ) ≥ (b m−M E b )1/(p−1) implies for all U ∈ E + , and assuming p > 1 in (f1 ), we get that kU kE = ρ < (b m/M Φ (U ) ≥ ω > 0

(3.18)

for all U ∈ E . On the other hand, we obtain by (f6 ) that +

Φ (U ) ≤

1 2

(kU k2E − λk−1 kU k22 ) ≤ 0

(3.19)

for all U ∈ E − . Now, given ε > 0, by the previous proposition there exists Rε > 0 such that N (U ) ≥

1

b bkU k22 − εkU k2E

2

for all U ∈ E with kU kE ≥ Rε . Since

kU k2E − b bkU k22 < kU k2E − λk−1 kU k22 ≤ 0, it follows that there exist mbb > 0 such that

kU k2E − b bkU k22 ≤ −mbb kU k2E , and taking 0 < ε < mbb we get that

Φ (U ) ≤ (−mbb + ε)kU k2E < 0 for all U ∈ E

−

⊕ Eλk , with kU kE ≥ Rε . 0

(3.20)

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P. Rabelo / Nonlinear Analysis 71 (2009) 2585–2599

Therefore, the estimates (3.18)–(3.20) show that the functional Φ exhibits the geometry required by the Generalized Mountain-Pass Theorem [8] and since this holds true when we replace the Palais–Smale condition by the Cerami condition (see [11]), we may conclude that Φ possesses a critical point b U ∈ E with Φ (b U ) > ω > 0, and in particular, b U 6= 0. Regularity and asymptotic behavior. We use a standard ‘‘bootstrap’’ argument to show that U is a strong solution of the problem (P). That is, each component of U is twice differentiable weakly in Rn and satisfies (P) almost everywhere. Indeed, let U ∈ W 1,2 (Rn , Rm ) satisfying

Z Rn

[∇ U · ∇ϕ + A(x)U · ϕ ] dx =

Z Rn

f (x, U ) · ϕ dx

for all ϕ ∈ Cc∞ (B2 ), where B2 = B(x0 , 2R) is a ball of radius 2R centered at x0 . Then, U is a weak solution of the equation

−∆U = h(x) in B2 ,

(3.21)

where h(x) = f (x, U (x)) − A(x)U (x). Letting 1 < p1 = 2∗ /p < 2∗ , follows from (2.2) and hypotheses (A4 ) and (A1 ) that

|h(x)|p1 ≤ |f (x, U (x)) + DU (x)|p1   ∗ ≤ C |U |p1 + |U |2 . Since W 1,2 (B2 , Rm ) ,→ L2 (B2 , Rm ) ,→ Lp1 (B2 , Rm ), ∗

we conclude that h ∈ Lp1 (B2 , Rm ). Now, if w is the Newtonian potential of h, it follows from [10, Theorem 9.9] that w ∈ W 2,p1 (B2 , Rm ) and

∆w = h(x) a.e. B2 .

(3.22)

Combining (3.21) and (3.22) we have that

Z

∇(U − w) · ∇ϕ dx = 0 B2

for all ϕ ∈ Cc∞ (B2 ). That is, U − w is a weak solution of ∆v = 0 in B2 . As U − w ∈ W 1,2 (B2 , Rm ), we can apply Weyl’s Lemma [12, Corollary 1.2.1] to conclude that U − w ∈ C ∞ (B2 , Rm ). Therefore, U ∈ W 2,p1 (B2 , Rm ). Since 1 < p < (n + 2)/(n − 2) there exist δ > 0 such that (n + 2)/(n − 2) = p(1 + δ). Hence p1 =

2n(1 + δ)

(n + 2)

.

Whereas W 2,p1 (B2 , Rm ) ,→ Lr1 (B2 , Rm ) with r1 = np1 /(n − 2p1 ), there exist p2 ∈ (p1 , r1 ) such that U ∈ W 2,p2 (B2 , Rm ). In fact, making p2 = r1 /p we have that r1 > p2 and as p2 p1

=

(n − 2)(1 + δ) > 1 + δ, n − 2 − 4δ

follows that p2 > p1 . Using the previous argument, we have that W 2,p1 (B2 , Rm ) ,→ Lr1 (B2 , Rm ) ,→ Lp2 (B2 , Rm ) and |h(x)|p2 ≤ C (|U |P2 + |U |r1 ), whence h ∈ Lp2 (B2 , Rm ) and U ∈ W 2,p2 (B2 , Rm ) with

 khkLp2 (B2 ,Rm ) ≤ C kU kLp2 (B2 ,Rm ) + kU kpLpp2 (B

2

 ,Rm )

.

Following in this fashion, we obtain an unbounded sequence pk+1 =

1 p



npk



n − 2pk

such that pk+1 /pk > 1 + δ and

 khkLpk+1 (B2 ,Rm ) ≤ C kU kLpk+1 (B2 ,Rm ) + kU kpLppk+1 (B

2

2 ,s

 ,R m )

.

Thus, U ∈ Wloc (Rn , Rm ) for all 2 ≤ s < +∞. By the Sobolev imbedding theorem, U ∈ C 1,γ (B2 , Rm ) with 0 < γ < 1 − n/s and s > n. Noticed that if the nonlinearities are of class C 1 or Hölder continuous, then U is a classical solution of the problem (P). By the interior Lp -estimates [10, Theorem 9.11] we have


kU kW 2,s (B1 ,Rm ) ≤ C kU kLs (B2 ,Rm ) + khkLs (B2 ,Rm ) ,

P. Rabelo / Nonlinear Analysis 71 (2009) 2585–2599

2599

where B1 = B(x0 , R). Hence,


kU kC 1,γ (B1 ,Rm ) ≤ C kU kLs (B2 ,Rm ) + kU kLsp (B2 ,Rm ) . Letting x0 → ∞, we conclude that kU kC 1,α (B1 ,Rm ) → 0. Multiplicity of solutions. As seen before in the application of the Mountain Pass Theorem, the conditions of growth (F1 )–(F4 ) and hypotheses (A1 )–(A4 ) on the potential, implies that the functional Φ is of class C 1 , Φ (0) = 0 and satisfies the Palais–Smale condition. Furthermore, as in the proof of the Theorem 1.1, the condition (F3 ) gives that on any subspace of finite dimension W ⊂ E, there is a R = R(W ) > 0 such that

Φ (u) |u∈∂ BR (0;W ) ≤ C1 R2 − C2 Rµ + C3 → −∞ as R → +∞. Similarly, we verify that there exists ρ, α > 0 such that Φ |∂ Bρ > α . Since Φ is even, we can apply the Symmetric Mountain Pass theorem to obtain an unbounded sequence the critical values of Φ under the assumptions of Theorem 1.1. To prove the existence of multiple solutions in the Theorem 1.2, we use a version of the Symmetric Mountain Pass theorem where the usual compactness condition of Palais–Smale is replaced by the compactness condition of Cerami (see [13, Proposition 2.2]). Acknowledgments This paper is a part of the author’s Ph.D. thesis at the UFPE department of mathematics and the author is greatly indebted to his thesis adviser Professor João Marcos do Ó for many useful discussions, suggestions, and comments. References [1] H. Buljan, T. Schwartz, M. Segev, M. Soljacic, D. Christoudoulides, Polychromatic partially spatially incoherent solitons in a non-instantaneous Kerr nonlinear medium, J. Opt. Soc. Amer. B 21 (2004) 397–404. [2] D. Christodoulides, E. Eugenieva, T. Coskun, M. Mitchell, M. Segev, Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media, Phys. Rev. E 63 (2001) 035–601. [3] S.M. Chang, C.S. Lin, T.C. Lin, W.W. Lin, Segregated nodal domains of two-dimensional multispecies Bose–Einstein condesates, Physica D 196 (2004) 341–361. [4] P.H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992) 272–291. [5] D.G. Costa, On a class of elliptic systems in Rn , Electron. J. Differential Equations 7 (1994) 1–14. [6] B. Sirakov, Existence and multiplicity of solutions of semi-linear elliptic equations in RN , Calc. Var. Partial Differential Equations 11 (2000) 119–142. [7] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381. [8] 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. [9] D.G. Costa, C.A. Magalhães, A variational approach to noncooperative elliptic systems, Nonlinear Anal. 25 (1995) 699–715. [10] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, 1983. [11] J.V. Gonçalves, O.H. Miyagaki, Existence of nontrivial solutions for semilinear elliptic equations at resonance, Houston J. Math. 16 (1990) 583–595. [12] J. Jost, Partial Differential Equations, in: Graduated Text in Mathematics, vol. 214, Springer, 2002. [13] G. Li, H. Zhou, Multiple solutions to p-Laplacian problems with asymptotic nonlinearity as up−1 at infinity, J. London Math. Soc. 65 (2002) 123–138.