Open Access

Multiplicity result for a critical elliptic system with concave-convex nonlinearities

Boundary Value Problems20132013:268

https://doi.org/10.1186/1687-2770-2013-268

Received: 10 April 2013

Accepted: 29 August 2013

Published: 6 December 2013

Abstract

We study the existence of multiple solutions of a strongly indefinite elliptic system involving the critical Sobolev exponent and concave-convex nonlinearities. By using a suitable version of the dual fountain theorem established in this paper, we prove the existence of infinitely many small energy solutions.

MSC:35A02, 35J50, 35J60.

Keywords

critical nonlinearityconcave-convex nonlinearitynoncooperative elliptic systemdual fountain theoremsmall energy solutions

1 Introduction

In this paper, we study the existence and multiplicity of solutions for the following elliptic system of noncooperative type:

where Ω is a bounded smooth domain in R N with N 3 , λ > 0 , and μ > 0 , 1 < p , q < 2 and α , β > 1 satisfy α + β = 2 , where 2 : = 2 N / N 2 denotes the critical Sobolev exponent. The solutions of ( S λ , μ ) are steady states of reaction-diffusion systems, which serve as a class of models with applications in physics, chemistry and biology (see, for instance, [1, 2]).

It is well known that weak solutions of ( S λ , μ ) are critical points of the following functional defined on the Hilbert space H 0 1 ( Ω ) × H 0 1 ( Ω ) :
J ( u , v ) : = Ω [ 1 2 ( | u | 2 | v | 2 ) λ p | u | p μ q | v | q + 1 α + β | u | α | v | β ] d x .
(1)
We call ( u , v ) a weak solution of ( S λ , μ ) if ( u , v ) H 0 1 ( Ω ) × H 0 1 ( Ω ) and
Ω [ u ϕ v φ λ | u | p 2 u ϕ μ | v | q 2 v φ + 1 α + β ( α | u | α 2 | v | β u ϕ + β | u | α | v | β 2 v φ ) ] d x = 0

for every ( ϕ , φ ) H 0 1 ( Ω ) × H 0 1 ( Ω ) .

The celebrated papers by Brezis and Nirenberg [3] and by Ambrosetti et al. [4] have inspired research on differential equations and systems with critical terms, and with concave-convex nonlinearities, respectively, see, for example, [512] and the many references therein. In recent years, more and more attention have been paid to the existence and multiplicity of solutions of the following elliptic system of cooperative type:

In [9], by extracting Palais-Smale sequences on the Nehari manifold, Hsu and Lin proved that if λ and μ are small, then ( P λ , μ ) has at least two solutions. Subsequently, their result was extended by Chen and Wu [13], who considered ( P λ , μ ) with sign-changing weight functions. See also [12, 14] and the references therein for some related results.

However, what can be said about the noncooperative case? To the best of our knowledge, there is no result on problem ( S λ , μ ), even in the subcritical case α + β < 2 . In addition to the combined effects of the concave and convex terms, two main difficulties arise when studying ( S λ , μ ). The first one is that the energy functional J above is strongly indefinite in the sense that it is neither bounded from above nor from below, even on subspaces of finite codimension. Therefore, the usual critical point theorems such as the mountain pass theorem cannot be used. Moreover, the method of the Nehari manifold, which is extensively used in the cooperative case does not apply in this situation. We refer to [15] for a unified approach on the method of the Nehari manifold. The second difficulty arises with the critical Sobolev exponent. Indeed, it is well known that the embedding H 0 1 ( Ω ) L 2 ( Ω ) is not compact because of the action of dilatations. Therefore, the energy functional is not expected to satisfy the Palais-Smale condition (see Definition 5 below ). Usually, the best we can expect is to find a bounded from above subset I R such that the functional satisfies the Palais-Smale condition at every level c I , see, for instance, [3, 9, 16, 17]. But surprisingly enough, we will show that in our case the functional J satisfies the Palais-Smale condition.

The main result of the paper is the following.

Theorem 1 Let λ , μ > 0 . Then problem ( S λ , μ ) has a sequence of solutions ( u k , v k ) such that J ( u k , v k ) < 0 and J ( u k , v k ) 0 as k .

Remark 2 Theorem 1 remains true in the subcritical case, i.e., if we assume that α + β > 2 if N = 1 , 2 or 2 < α + β < 2 if N = 3 . In this case, the proof of Theorem 1 is considerably simplified, since the embedding H 0 1 ( Ω ) L α + β ( Ω ) is now compact.

The paper is organized as follows. In order to prove Theorem 1, we need a new critical point theorem for strongly indefinite functionals, which will be provided in Section 2. This critical point theorem generalizes the dual fountain theorem of Bartsch and Willem, and is not based on any reduction method. It should be noted that problem ( S λ , μ ) does not fit into the framework of the theorem proved in [18], which is only suitable for finding large energy solutions. Finally the proof of our main result is presented in Section 3.

2 Critical point theory

Let Y be a closed subspace of a separable Hilbert space X endowed with the inner product ( ) and the associated norm . We denote by P : X Y and Q : X Z : = Y the orthogonal projections.

We fix an orthonormal basis ( e j ) j 0 of Y, and we consider on X = Y Z the τ-topology introduced by Kryszewski and Szulkin in [19]; that is, the topology associated to the following norm
u : = max ( j = 0 1 2 j + 1 | ( P u , e j ) | , Q u ) , u X .
Clearly Q u u u . Moreover, if ( u n ) is a bounded sequence in X, then
u n τ u P u n P u and Q u n Q u .

Definition 3 Let Φ C 1 ( X , R ) and a , b R , a < b . We say that Φ is τ-weak sequentially compact in Φ 1 ( [ a , b ] ) if for every sequence u n which τ-converges to u in Φ 1 ( [ a , b ] ) , there is a subsequence ( u n k ) such that Φ ( u n k ) Φ ( u ) .

Lemma 4 (Deformation lemma)

Let φ C 1 ( X , R ) be an even functional which is τ-upper semicontinuous, and such that φ is τ-weak sequentially compact in φ 1 ( [ a , b ] ) for any a , b R . Let S X with S = S , and c R , ε , θ > 0 such that
u φ 1 ( [ c 2 ε , c + 2 ε ] ) S 2 θ , φ ( u ) 8 ε θ .
Then there exists η C ( [ 0 , 1 ] × φ c + 2 ε , X ) such that
  1. (i)

    η ( t , u ) = u if t = 0 or if u φ 1 ( [ c 2 ε , c + 2 ε ] ) S 2 θ ,

     
  2. (ii)

    η ( 1 , φ c + ε S ) φ c ε ,

     
  3. (iii)

    η ( t , u ) u θ 2 u φ c + 2 ε , t [ 0 , 1 ] ,

     
  4. (iv)

    φ ( η ( , u ) ) is non increasing, u φ c + 2 ε ,

     
  5. (v)

    Each point ( t , u ) [ 0 , 1 ] × φ c + 2 ε has a τ-neighborhood N ( t , u ) such that { v η ( s , v ) | ( s , v ) N ( t , u ) ( [ 0 , 1 ] × φ c + 2 ε ) } is contained in a finite-dimensional subspace of X,

     
  6. (vi)

    η is τ-continuous,

     
  7. (vii)

    η ( t , ) is odd t [ 0 , 1 ] ,

     

where S α : = { u X | dist ( u , S ) α } α > 0 and φ a : = { u X | φ ( u ) a } a R .

Proof We define
w ( v ) : = 2 φ ( v ) 2 φ ( v ) , v φ 1 ( [ c 2 ε , c + 2 ε ] ) .

We claim that for every v φ 1 ( [ c 2 ε , c + 2 ε ] ) , there exists a τ-open neighborhood N v of v such that ( φ ( u ) , w ( v ) ) > 1 u N v ( N v can be chosen to be symmetric, that is, N v = N v ). In fact, if this is not true, then we can find a sequence ( v n ) φ 1 ( [ c 2 ε , c + 2 ε ] ) which τ-converges to an element v φ 1 ( [ c 2 ε , c + 2 ε ] ) and such that ( φ ( v n ) , w ( v ) ) 1 . Since φ is τ-weak sequentially compact, there is a subsequence ( v n k ) such that φ ( v n k ) φ ( v ) , and this implies that ( φ ( v ) , w ( v ) ) 1 , which is in contradiction with the definition of w.

The rest of the proof follows the same lines as the proof of Lemma 8 in [18] with G = Z 2 . □

We introduce the following notations:
Y k : = j = k R e j ¯ , Z k : = ( j = 0 k R e j ) Z , B k : = { u Y k | u σ k } , N k : = { u Z k | u = s k } , with  0 < s k < σ k , k 2 .

Before we state and prove our critical point theorem, we recall the following definition.

Definition 5 A functional Φ C 1 ( X , R ) is said to satisfy the Palais-Smale condition (resp. ( P S ) c condition ) if every sequence ( u n ) X such that ( Φ ( u n ) ) is bounded (resp. Φ ( u n ) c ) and Φ ( u n ) 0 , has a convergent subsequence.

Theorem 6 (Generalized dual fountain theorem)

Let Φ C 1 ( X , R ) be an even functional which is τ-lower semicontinuous and such that Φ is τ-weak sequentially compact in Φ 1 ( [ a , b ] ) for any a , b R . If, for every k k 0 , there exist σ k > s k > 0 such that

(B1) a k : = inf u Y k u = σ k Φ ( u ) 0 ,

(B2) b k : = sup u Z k u = s k Φ ( u ) < 0 ,

(B3) d k : = inf u Y k u σ k Φ ( u ) 0 , k ,

(B4) Φ satisfies the ( P S ) c condition c [ d k 0 , 0 [ .

Then Φ has a sequence of critical points ( u k ) such that Φ ( u k ) < 0 and Φ ( u k ) 0 as k .

Proof Let Γ k be the set of maps γ : B k X such that
  1. (a)

    γ is odd, τ-continuous and γ | B k = id ,

     
  2. (b)

    each u int ( B k ) has a τ-neighborhood N u in Y k such that ( id γ ) ( N u int ( B k ) ) is contained in a finite dimensional subspace of X,

     
  3. (c)

    Φ ( γ ( u ) ) Φ ( u ) u B k .

     
Define
c k : = sup γ Γ k inf u B k Φ ( γ ( u ) ) .

It follows from the definitions that d k c k . Lemma 10 in [18] implies that γ ( B k ) N k γ Γ k , which in turn implies that c k b k .

Let ε ] 0 , a k c k 2 [ , θ > 0 , and let γ Γ k such that
c k ε inf u B k Φ ( γ ( u ) ) .
(2)
We claim that
u Φ 1 ( [ c k 2 ε , c k + 2 ε ] ) ( γ ( B k ) ) 2 θ such that  Φ ( u ) 8 ε θ .
(3)
Hence, there exists a sequence ( u n ) X such that
Φ ( u n ) c k and Φ ( u n ) 0 , n .

The conclusion of Theorem 6 then follows from (B4) and (B3).

To complete the proof of Theorem 6, it remains to show that (3) holds.

We proceed by contradiction by assuming that (3) does not hold, and then, we apply Lemma 4 with φ = Φ , c = c k and S = γ ( B k ) . We may assume that c k + 2 ε < a k . Next, by following the proof of Theorem 11 in [18], one can easily verify that the map β defined on B k by β ( u ) : = η ( 1 , γ ( u ) ) belongs to Γ k . Now, (2) and (ii) of Lemma 4 imply that
inf u B k Φ ( β ( u ) ) c k + ε ,

which contradicts the definition of c k . □

3 Proof of the main results

In this section, we denote by | u | p the usual L p ( Ω ) norm.

We consider the Sobolev space H 0 1 ( Ω ) endowed with the norm u = | u | 2 and X : = H 0 1 ( Ω ) × H 0 1 ( Ω ) with the product norm ( u , v ) = ( u 2 + v 2 ) 1 2 .

The functional J defined by (1) then reads as follows:
J ( u , v ) : = 1 2 u 2 1 2 v 2 λ p | u | p p μ q | v | q q + 1 α + β Ω | u | α | v | β d x .
(4)
A standard argument shows that J is of class C 1 on X and
J ( u , v ) , ( ϕ , φ ) = Ω [ u ϕ v φ λ | u | p 2 u ϕ μ | v | q 2 v φ + 1 α + β ( α | u | α 2 | v | β u ϕ + β | u | α | v | β 2 v φ ) ] d x .
(5)

Lemma 7 Let ( u n , v n ) X such that ( u n , v n ) ( u , v ) in X. Then there exists a subsequence ( u n k , v n k ) such that J ( u n k , v n k ) J ( u , v ) .

Proof Let ( ϕ , φ ) X . Using (5), we have
J ( u n , v n ) J ( u , v ) , ( ϕ , φ ) = Ω [ ( u n u ) ϕ ( v n v ) φ λ ( | u n | p 2 u n | u | p 2 u ) ϕ μ ( | v n | q 2 v n | v | q 2 v ) φ + α α + β ( | u n | α 2 u n | v n | β | u | α 2 u | v | β ) ϕ + β α + β ( | u n | α | v n | β 2 v n | u | α | v | β 2 v ) φ ] d x .
It is clear that
Ω ( u n u ) ϕ d x 0 and Ω ( v n v ) φ d x 0 .
Now up to a subsequence we have
u n u and v n v a.e. on  Ω .

By the Sobolev embedding theorem ( u n ) and ( v n ) are bounded in L p ( Ω ) , L q ( Ω ) and L α + β ( Ω ) . So ( | u n | p 2 u n ) (resp. ( | v n | q 2 v n ) ) is bounded in L p p 1 ( Ω ) (resp. in L q q 1 ( Ω ) ), ( | u n | α 2 u n | v n | β ) and ( | u n | α | v n | β 2 v n ) are bounded in L α + β α + β 1 ( Ω ) .

By Theorem 10.36 in [20], we have
| u n | p 2 u n | u | p 2 u in  L p p 1 ( Ω ) , | v n | q 2 v n | v | q 2 v in  L q q 1 ( Ω ) , | u n | α 2 u n | v n | β | u | α 2 u | v | β and | u n | α | v n | β 2 v n | u | α | v | β 2 v in  L α + β α + β 1 ( Ω ) .
Since ϕ , φ H 0 1 ( Ω ) L p ( Ω ) L q ( Ω ) L α + β ( Ω ) , it follows that
Ω ( | u n | p 2 u n | u | p 2 u ) ϕ d x 0 , Ω ( | v n | q 2 v n | v | q 2 v ) φ d x 0 , Ω ( | u n | α 2 u n | v n | β | u | α 2 u | v | β ) ϕ d x 0 , Ω ( | u n | α | v n | β 2 v n | u | α | v | β 2 v ) φ d x 0 ,

and J ( u n , v n ) J ( u , v ) , ( ϕ , φ ) 0 for every ( ϕ , φ ) X . Hence, J ( u n , v n ) J ( u , v ) . □

Lemma 8 The functional J satisfies the Palais-Smale condition.

Proof Let ( u n , v n ) X such that
d : = sup n | J ( u n , v n ) | < and J ( u n , v n ) 0 , n .
We deduce from (5) that
J ( u n , v n ) , ( u n , 0 ) = u n 2 λ | u n | p p + α α + β Ω | u n | α | v n | β d x u n 2 λ | u n | p p .
Hence for n big enough, we have
u n 2 λ | u n | p p u n ,

which implies, since p < 2 , that ( u n ) is bounded.

On the other hand, by using (4) and (5), we obtain
J ( u n , v n ) + 1 α + β J ( u n , v n ) , ( u n , v n ) = ( 1 α + β 1 2 ) u n 2 + ( 1 2 1 α + β ) v n 2 + λ ( 1 p 1 α + β ) | u n | p p + μ ( 1 q 1 α + β ) | v n | q q .
Since ( u n ) is bounded, q < 2 < α + β and
J ( u n , v n ) + 1 α + β J ( u n , v n ) , ( u n , v n ) d + ( u n , v n )

for n big enough, we easily deduce that ( v n ) is bounded.

Consequently, we have, up to a subsequence,
( u n , v n ) ( u , v ) in  X ,
(6)
u n u in  L p ( Ω ) , v n v in  L q ( Ω ) , u n u , v n v a.e. on  Ω , J ( u n , v n ) J ( u , v ) ( by Lemma  7 ) .
(7)
On the other hand, J ( u n , v n ) 0 and equation (6) imply that J ( u , v ) = 0 and that
J ( u n , v n ) , ( u n , 0 ) J ( u , v ) , ( u , 0 ) 0 , J ( u n , v n ) , ( 0 , v n ) J ( u , v ) , ( 0 , v ) 0 .
Now, because of equation (7), it follows that
u n 2 u 2 + α α + β Ω ( | u n | α | v n | β | u | α | v | β ) d x 0 , v n 2 + v 2 + β α + β Ω ( | u n | α | v n | β | u | α | v | β ) d x 0 .
Let us combine the preceding equations in order to cancel the integrals. We then get
β ( u n 2 u 2 ) + α ( v n 2 v 2 ) 0 .
But since equation (6) leads to
u n 2 = u n u 2 + u 2 + ( 1 ) , v n 2 = v n v 2 + v 2 + ( 1 ) ,
it is then easy to conclude that
β u n u 2 + α v n v 2 0 .

Hence ( u n , v n ) ( u , v ) . □

Now, we select an orthonormal basis ( e j ) of H 0 1 ( Ω ) , and we consider the τ-topology on X = Y Z , where Y and Z are defined by
Y : = H 0 1 ( Ω ) × { 0 } and Z : = { 0 } × H 0 1 ( Ω ) .

Lemma 9 J is τ-lower semicontinuous, and J is τ-weak sequentially compact in J 1 ( [ a , b ] ) for any a , b R .

Proof
  1. 1.
    Let ( u n , v n ) X and C R such that ( u n , v n ) τ ( u , v ) in X and J ( u n , v n ) C . By the definition of τ, ( v n ) converges strongly to v in H 0 1 ( Ω ) . Clearly, we have
    J ( u n , v n ) 1 2 u n 2 1 2 v n 2 λ p | u n | p p μ q | v n | q q ,
    which implies, since ( v n ) is bounded and p < 2 , that ( u n ) is bounded. Up to a subsequence, we may assume that u n u in L p ( Ω ) and v n v in L q ( Ω ) , u n ( x ) u ( x ) and v n ( x ) v ( x ) for almost every x Ω . By using Fatou’s lemma and the weak lower semicontinuity of the norm , we deduce from the inequality
    C J ( u n , v n ) = 1 2 u n 2 1 2 v n 2 λ p | u n | p p μ q | v n | q q + 1 α + β Ω | u n | α | v n | β d x

    that C J ( u , v ) .

     
  2. 2.

    Assume that ( u n , v n ) τ ( u , v ) in J 1 ( [ a , b ] ) , a , b R . The same argument as above shows that ( u n , v n ) is bounded, and then ( u n , v n ) ( u , v ) . It follows from Lemma 7 that J is τ-weak sequentially compact in J 1 ( [ a , b ] ) .

     

 □

Proof of Theorem 1 We recall that
Y k : = j = k R e j ¯ × { 0 } and Z k : = ( j = 0 k R e j × { 0 } ) Z .
Let ( u , 0 ) Y k . Then, since λ > 0 , we have
J ( u , 0 ) = 1 2 u 2 λ p | u | p p 1 2 u 2 λ p θ k p u p ,
where
θ k = sup w j = k R e j ¯ w = 1 | w | p .
Therefore, for every ( u , 0 ) Y k such that u = σ k : = 2 ( 2 λ p θ k p ) 1 2 p , we have
J ( u , 0 ) ( λ p θ k p ) 2 2 p 2 p 2 p ( 2 2 2 p ) > 0 .
On the other hand, it is clear that
1 2 u 2 J ( u , 0 ) λ p θ k p u p .
Hence, for every ( u , 0 ) Y k such that u σ k , we have
1 2 σ k 2 J ( u , 0 ) λ p θ k p σ k p .

We know by [21] that θ k 0 as k , hence we deduce that J ( u , 0 ) 0 as k .

We have then proved that assumptions (B1) and (B3) of Theorem 6 are satisfied.

Let ( u , v ) Z k . Since the norms and | | p are equivalent on j = 0 k R e j , μ > 0 and H 0 1 ( Ω ) continuously embeds into L α + β ( Ω ) , we have
J ( u , v ) 1 2 u 2 λ c 1 p u p + c α + β u α + β 1 2 v 2 + c α + β v α + β ,

where c 1 > 0 and c > 0 are constants. It is then easy to verify, since λ > 0 , that assumption (B2) of Theorem 6 is satisfied for s k small enough.

By using Lemmas 8, 9, we can apply Theorem 6 and get the desired result. □

Declarations

Acknowledgements

The authors are grateful to the anonymous referees for their careful reading of the paper and for helpful comments. This work was funded by a grant from the Natural Sciences and Engineering Research Council of Canada.

Authors’ Affiliations

(1)
Département de Mathématiques, Université de Sherbrooke
(2)
Mathematics and Computer Science Department, Laurentian University

References

  1. Murray JD: Mathematical Biology. Springer, Berlin; 1993.View ArticleMATHGoogle Scholar
  2. Smoller J: Shock Waves and Reaction-Diffusion Equations. Springer, Berlin; 1983.View ArticleMATHGoogle Scholar
  3. Brezis H, Nirenberg L: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 1983, 36: 437-477. 10.1002/cpa.3160360405MathSciNetView ArticleMATHGoogle Scholar
  4. Ambrosetti A, Brezis H, Cerami G: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 1994, 122(2):519-543. 10.1006/jfan.1994.1078MathSciNetView ArticleMATHGoogle Scholar
  5. Bartsch T, Willem M: On an elliptic equation with concave and convex nonlinearities. Proc. Am. Math. Soc. 1995, 123(11):3555-3561. 10.1090/S0002-9939-1995-1301008-2MathSciNetView ArticleMATHGoogle Scholar
  6. Cheng X, Ma S: Existence of three nontrivial solutions for elliptic systems with critical exponents and weights. Nonlinear Anal. 2008, 69: 3537-3548. 10.1016/j.na.2007.09.040MathSciNetView ArticleMATHGoogle Scholar
  7. Wu TF: On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function. J. Math. Anal. Appl. 2006, 318(1):253-270. 10.1016/j.jmaa.2005.05.057MathSciNetView ArticleMATHGoogle Scholar
  8. Batkam, CJ, Colin, F: The effects of a Concave-Convex nonlinearity in some noncooperative elliptic systems. Ann. Mat. Pura Appl. Published online 21 April 2013 (to appear)Google Scholar
  9. Hsu TS, Lin HL: Multiple positive solutions for a critical elliptic system with concave-convex nonlinearities. Proc. R. Soc. Edinb., Sect. A, Math. 2009, 139(6):1163-1177. 10.1017/S0308210508000875MathSciNetView ArticleMATHGoogle Scholar
  10. Shen Y, Zhang J: Multiplicity of positive solutions for a semilinear p-Laplacian system with Sobolev critical exponent. Nonlinear Anal. 2011, 74(4):1019-1030. 10.1016/j.na.2010.06.066MathSciNetView ArticleMATHGoogle Scholar
  11. Han P: Multiple positive solutions of nonhomogeneous elliptic systems involving critical Sobolev exponents. Nonlinear Anal. 2006, 64(4):869-886. 10.1016/j.na.2005.04.053MathSciNetView ArticleMATHGoogle Scholar
  12. Wu TF: The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions. Nonlinear Anal. 2008, 68: 1733-1745. 10.1016/j.na.2007.01.004MathSciNetView ArticleMATHGoogle Scholar
  13. Chen CY, Wu TF: The Nehari manifold for indefinite semilinear elliptic systems involving critical exponent. Appl. Math. Comput. 2012, 218: 10817-10828. 10.1016/j.amc.2012.04.026MathSciNetView ArticleMATHGoogle Scholar
  14. Hsu TS: Multiple positive solutions for a critical quasilinear elliptic system with concave-convex nonlinearities. Nonlinear Anal. 2009, 71: 2688-2698. 10.1016/j.na.2009.01.110MathSciNetView ArticleMATHGoogle Scholar
  15. Szulkin A, Weth T: The method of Nehari manifold. In Handbook of Nonconvex Analysis and Applications. Int. Press, Somerville; 2010:597-632.Google Scholar
  16. Bouchekif M, Nasri Y: On a singular elliptic system at resonance. Ann. Mat. Pura Appl. 2010, 189(2):227-240. 10.1007/s10231-009-0106-9MathSciNetView ArticleMATHGoogle Scholar
  17. Chabrowski J: The Neumann problem for semilinear elliptic equations with critical Sobolev exponent. Milan J. Math. 2007, 75: 197-224. 10.1007/s00032-006-0065-1MathSciNetView ArticleMATHGoogle Scholar
  18. Batkam CJ, Colin F: Generalized fountain theorem and application to strongly indefinite semilinear problems. J. Math. Anal. Appl. 2013, 405: 438-452. 10.1016/j.jmaa.2013.04.018MathSciNetView ArticleMATHGoogle Scholar
  19. Kryszewski W, Szulkin A: Generalized linking theorem with an application to a semilinear Schrödinger equation. Adv. Differ. Equ. 1998, 3: 441-472.MathSciNetMATHGoogle Scholar
  20. Willem M: Analyse Harmonique Réelle. Hermann, Paris; 1995.MATHGoogle Scholar
  21. Willem M: Minimax Theorems. Birkhauser, Boston; 1996.View ArticleMATHGoogle Scholar

Copyright

© Batkam and Colin; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.