In this paper, the existence of at least one nontrivial solution for a class of semilinear elliptic equations on is established by using the linking methods.
Keywords:Schrödinger equation; subcritical exponent; local linking
In this paper we consider the question of the existence of solutions for a class of semilinear equations of the form
where is a parameter and the nonlinearity is asymptotically linear, i.e.,
for some and . In case this equation is considered in a bounded domain (with, say, the Dirichlet boundary condition), there is a large amount of literature on existence and multiplicity results, with the case of resonance being of particular interest (see [1-3]). We recall that the problem is said to be at resonance if , where denotes the spectrum of S, the ‘asymptotic linearization’ of the problem. In other words, is the operator given by
On the other hand, a systematic study of such asymptotically linear problems set in unbounded domains or the whole space is more recent and presents a number of mathematical difficulties (see [4,5]). As an example, we note that in the case of problem ( ), the asymptotic linearization operator S (now defined on ) has a much more complicated spectrum (including an essential part ), which in turn makes the study of this problem more challenging. In , motivated by the paper , Tehrani and Costa studied the existence of positive solutions to ( ) by using the mountain pass theorem if satisfies some strong asymptotically linear conditions. Comparing with previous paper , in , Tehrani obtained the existence of a (possibly sign-changing) solution for problem ( ) under essentially condition (1.1) only. In fact, he proved the following.
Let and assume that
(G) for every , there exists such that
If or and , then ( ) has a solution in .
Now, one naturally asks: Are there nontrivial solutions for problem ( ) if in the above theorem? Obviously, this case is resonance. But, this problem is not easy because we face the difficulties of verifying that the energy functional satisfies the (PS) condition if we still follow the idea of . Here, there is still an interesting problem: Are there nontrivial solutions for problem ( ) if and (in Theorem 1.0) is more generalized superlinear? We will answer the above problems affirmatively by using Li and Willem’s local linking methods (see ).
Next, we recall a few basic facts in the theory of Schrödinger operators which are relevant to our discussion (see ).
1. Since , one has .
2. The bottom of the spectrum of the operator S is given by
Therefore we clearly have . If , then by using the concentration compactness principle of Lions, one shows that Λ is the principle eigenvalue of S with a positive eigenfunction :
3. The spectrum of S in , namely , is at most a countable set, which we denote by
where each is an isolated eigenvalue of S of the finite multiplicity. Let denote the eigenspace of S corresponding to the eigenvalue .
Now, we state our main results. In this paper, we always assume that and . The conditions imposed on (see Theorem 1.0) are as follows:
(H1) , and there are constants such that
(H2) , , uniformly on ;
(H3) uniformly on ;
(H4) There is a constant such that for all and ,
(H5) For some , either
(H6) for every .
Theorem 1.1Assume that conditions (H1)-(H4) hold. If −λis an eigenvalue of , assume also that (H5) and (H6) hold. Then the problem ( ) has at least one nontrivial solution.
Remark 1.1 It follows from the condition (H3) that our nonlinearity does not satisfy the classical condition of Ambrosetti and Rabinowitz:
(AR) There is such that for all and .
We also consider asymptotically quadratic functions. We assume that:
(H7) For every , there exists such that
Theorem 1.2Assume that conditions (H2), (H6), (H7) and one of the following conditions hold:
(A1) , ;
(A2) , for some ,
(A3) , for some ,
Then problem ( ) has at least one nontrivial solution.
Let X be a Banach space with a direct sum decomposition
Consider two sequences of subspaces
For every multi-index , let . We know that
A sequence is admissible if, for every , there is such that . For every , we denote by the function I restricted .
Definition 2.1 Let I be locally Lipschitz on X and . The functional I satisfies the condition if every sequence such that is admissible and
contains a subsequence which converges to a critical point of I.
Definition 2.2 Let I be locally Lipschitz on X and . The functional I satisfies the condition if every sequence such that is admissible and
contains a subsequence which converges to a critical point of I.
Remark 2.1 1. The condition implies the condition for every .
2. When the sequence is bounded, then the sequence is a sequence (see ).
3. Without loss of generality, we assume that the norm in X satisfies
Definition 2.3 Let X be a Banach space with a direct sum decomposition
The function has a local linking at 0, with respect to if, for some ,
Lemma 2.1 (see )
Suppose that satisfies the following assumptions:
(B1) Ihas a local linking at 0 and ;
(B2) Isatisfies ;
(B3) Imaps bounded sets into bounded sets;
(B4) For every , , , . ThenIhas at least two critical points.
Remark 2.2 Assume that I satisfies the condition. Then this theorem still holds.
Let X be a real Hilbert space and let . The gradient of I has the form
where A is a bounded self-adjoint operator, 0 is not the essential spectrum of A, and B is a nonlinear compact mapping.
We assume that there exist an orthogonal decomposition,
and two sequences of finite-dimensional subspaces,
For every multi-index , we denote by the space
by the orthogonal projector onto , and by the Morse index of a self-adjoint operator L.
Lemma 2.2 (see )
Isatisfies the following assumptions:
(i) Ihas a local linking at 0 with respect to ;
(ii) There exists a compact self-adjoint operator such that
(iii) is invertible;
(vi) For infinitely many multiple-indices ,
ThenIhas at least two critical points.
3 The proof of main results
Proof of Theorem 1.1 (1) We shall apply Lemma 2.1 to the functional
defined on . We consider only the case , and
Then other case is similar and simple.
Let be a finite dimensional space spanned by the eigenfunctions corresponding to negative eigenvalues of and let be its orthogonal complement in X. Choose a Hilbertian basis ( ) for X and define
By the condition (H1) and Sobolev inequalities, it is easy to see that the functional I belongs to and maps bounded sets to bounded sets.
(2) We claim that I has a local linking at 0 with respect to . Decompose into when , . Also, set , , , .
For the convenience of our proof, we state some facts for the norm of the whole space X. It is well known that there is an equivalent norm on such that
By the equivalence of norm in the finite-dimensional space, there exists such that
It follows from (H1) and (H2) that for any , there exists such that
Hence, we obtain
where , is a constant and hence, for small enough,
Let be such that and let
From (3.2), we have
for all and . On the one hand, one has for all . Hence, from (H5), we obtain
On the other hand, we have
for all . It follows from (3.3) that
for all and all with , which implies that
where is a constant. Hence, there exist positive constants , and such that
for all with , which implies that
for small enough.
(3) We claim that I satisfies . Consider a sequence such that is admissible and
Let . Up to a subsequence, we have
If , we choose a sequence such that
For any , let . Now, we claim that
Let ; for , then,
Since , we may fix r large enough such that
for all n. Moreover, by (H6), there exists such that
for all n. Finally, since in for , we can use (H1) again to derive
for n large enough. Combining the above three formulas, our claim holds.
So, for n large enough, , we have
where ϵ is a small enough constant.
That is, . Now, , , we know that and
Therefore, using (H4), we have
This contradicts (3.5).
If , then the set has a positive Lebesgue measure. For , we have . Hence, by (H3), we have
From (3.4), we obtain
By (3.8), the right-hand side of (3.9) . This is a contradiction.
In any case, we obtain a contradiction. Therefore, is bounded.
Next, we denote as and prove contains a convergent subsequence.
In fact, we know that is bounded in X. Passing to a subsequence, we may assume that in X. In order to establish strong convergence, it suffices to show that
By the condition (H6) and , we can similarly conclude it according to the above proof of our claim.
Finally, we claim that for every ,
By (H2) and (H3), there exist large enough M and some positive constant T such that
So, for any , we have
Hence, our claim holds. □
Proof of Theorem 1.2 We omit the proof which depends on Lemma 2.2 and is similar to the preceding one since our result is a variant of Ding Yanheng’s Theorem 1.2 (see ). □
The authors declare that they have no competing interests.
The authors read and approved the final manuscript.
The authors would like to thank the referees for valuable comments and suggestions in improving this paper. This work was supported by the National NSF (Grant No. 10671156) of China.
Bartolo, P, Benci, V, Fortunato, D: Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity. Nonlinear Anal.. 7, 981–1012 (1983). Publisher Full Text
Capozzi, A, Lupo, D, Solimini, S: On the existence of a nontrivial solution to nonlinear problems at resonance. Nonlinear Anal.. 13, 151–163 (1989). Publisher Full Text
Costa, DG, Tehrani, HT: On a class of asymptotically linear elliptic problems in . J. Differ. Equ.. 173, 470–494 (2001). Publisher Full Text
Stuart, CA, Zhou, HS: Applying the mountain pass theorem to an asymptotically linear elliptic equation on . Commun. Partial Differ. Equ.. 24, 1731–1758 (1999). Publisher Full Text
Tehrani, HT: A note on asymptotically linear elliptic problems in . J. Math. Anal. Appl.. 271, 546–554 (2002). Publisher Full Text
Li, SJ, Willem, M: Applications of local linking to critical point theory. J. Math. Anal. Appl.. 189, 6–32 (1995). Publisher Full Text
Jeanjean, L, Tanaka, K: A positive solution for a nonlinear Schrödinger equation on . Indiana Univ. Math. J.. 54, 443–464 (2005). Publisher Full Text
Teng, KM: Existence and multiplicity results for some elliptic systems with discontinuous nonlinearities. Nonlinear Anal.. 75, 2975–2987 (2012). Publisher Full Text
Ding, YH: Some existence results of solutions for the semilinear elliptic equations on . J. Differ. Equ.. 119, 401–425 (1995). Publisher Full Text