- Open Access
Existence of solutions for semilinear elliptic equations on
© Pei and Zhang; licensee Springer. 2013
Received: 28 March 2013
Accepted: 8 June 2013
Published: 9 July 2013
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.
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.
Theorem 1.0 
- (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 ).
Since , one has .
- 2.The bottom of the spectrum of the operator S is given by
- 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:
(H2) , , uniformly on ;
(H3) uniformly on ;
(H6) for every .
Theorem 1.1 Assume that conditions (H1)-(H4) hold. If −λ is 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:
Theorem 1.2 Assume that conditions (H2), (H6), (H7) and one of the following conditions hold:
(A1) , ;
Then problem () has at least one nontrivial solution.
A sequence is admissible if, for every , there is such that . For every , we denote by the function I restricted .
contains a subsequence which converges to a critical point of I.
contains a subsequence which converges to a critical point of I.
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
Lemma 2.1 (see )
Suppose that satisfies the following assumptions:
(B1) I has a local linking at 0 and ;
(B2) I satisfies ;
(B3) I maps bounded sets into bounded sets;
(B4) For every , , , . Then I has at least two critical points.
Remark 2.2 Assume that I satisfies the condition. Then this theorem still holds.
where A is a bounded self-adjoint operator, 0 is not the essential spectrum of A, and B is a nonlinear compact mapping.
by the orthogonal projector onto , and by the Morse index of a self-adjoint operator L.
Lemma 2.2 (see )
I has a local linking at 0 with respect to ;
- (ii)There exists a compact self-adjoint operator such that
- (vi)For infinitely many multiple-indices ,
Then I has at least two critical points.
3 The proof of main results
Then other case is similar and simple.
We claim that I has a local linking at 0 with respect to . Decompose into when , . Also, set , , , .
- (3)We claim that I satisfies . Consider a sequence such that is admissible and(3.4)
for n large enough. Combining the above three formulas, our claim holds.
where ϵ is a small enough constant.
This contradicts (3.5).
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.
By the condition (H6) and , we can similarly conclude it according to the above proof of our claim.
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 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.
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