Multiple solutions to nonlinear Schrödinger equations with critical growth
© Liu and Zhao; licensee Springer 2013
Received: 20 March 2013
Accepted: 24 May 2013
Published: 4 September 2013
In 2000, Cingolani and Lazzo (J. Differ. Equ. 160:118-138, 2000) studied nonlinear Schrödinger equations with competing potential functions and considered only the subcritical growth. They related the number of solutions with the topology of the global minima set of a suitable ground energy function. In the present paper, we establish these results in the critical case. In particular, we remove the condition , which is a key condition in their paper. In the proofs we apply variational methods and Ljusternik-Schnirelmann theory.
Keywordsnonlinear Schrödinger equations critical growth variational methods
1 Introduction and main result
A considerable amount of work has been devoted to investigating solutions of (1.3). The existence, multiplicity and qualitative property of such solutions have been extensively studied. For single interior spikes solutions in the whole space , please see [1–9]etc. For multiple interior spikes, please see [10, 11]etc. For single boundary spike solutions with Neumann boundary condition, please see [6, 12–15]etc. For multiple boundary spikes, please see [16–18]etc. In particular, Wang and Zeng  studied the existence and concentration behavior of solutions for NLS with competing potential functions. Cingolani and Lazzo in  obtained the multiple solutions for the similar equation. In those papers only the subcritical growth was considered. In the present paper, we complete these studies by considering a class of nonlinearities with the critical growth. In particular, we remove the condition , which is a key condition in .
where if , and if . satisfies
(f1) for each ;
(f5) the function is strictly increasing in for any .
Our main results are the following theorem.
Theorem 1.1 Let . Suppose that f satisfies (f1)-(f5), V is a continuous function in and satisfies . Then when ε is sufficiently small, the problem (1.4) has at least distinct nontrivial solutions.
Here denotes the Ljusternik-Schnirelmann category of Σ in . By definition (e.g., ), the category of A with respect to M, denoted by , is the least integer k such that , with () closed and contractible in M. We set and if there are no integers with the above property. We will use the notation for .
We will also prove that if u is a critical point of satisfying , then u cannot change sign. Hence we obtain at least nontrivial critical points of .
The paper is organized as follows. In Section 2, we collect some notations and preliminaries. A compactness result is given in Section 3, which is a key step in our proof. Finally, in Section 4, we prove Theorem 1.1.
2 Notations and preliminaries
Hence, the weak solutions of (1.4) are exactly the critical points of .
3 Compactness result
Proposition 3.1 Let as . Assume that satisfies as . Then uniformly in , there exist a subsequence of (still denoted by ), and such that . Furthermore, converges strongly in to w, the positive ground state solution of equation (2.2).
Hence , from which it follows that in .
By the concentration-compactness lemma , there exists a subsequence of (denoted in the same way) satisfying one of the three following possibilities.
Hence , contradicting .
where as .
Contradiction! Thus dichotomy does not occur.
for some positive constants , independent of δ, which implies , contrary to (3.8).
i.e., w is the ground state solution of (2.2) in view of (3.13). The proof of Proposition 3.1 is complete. □
4 Proof of Theorem 1.1
Proposition 4.1 Suppose f satisfies (f2)-(f4). Then satisfies the -condition for all , that is, every sequence in such that , , as , possesses a convergent subsequence.
Letting , we get , so either which contradicts (4.5) or . □
which contradicts (4.12). Thus, up to a subsequence, .
Thus (4.7) is proved. □
uniformly for .
By Proposition 3.1, converges strongly in to w, which is a positive ground state solution of equation (2.2). Thanks to the exponential decay of w (see (2.4)), we obtain as . This completes the proof of Lemma 4.3. □
Proof of Theorem 1.1 By Proposition 4.1, satisfies the -condition for all . Now let us choose a function such that as and such that is not a critical level for . For such , let us introduce the set as in (4.6). Then the standard Ljusternik-Schnirelmann theory implies that has at least critical points on (also see ).
which is a contradiction. Therefore there exist at least nonzero critical points of and thus solutions of equation (1.4). □
This work was partially supported by the National Natural Science Foundation of China (11261052). The authors are grateful to Prof. Guowei Dai for pointing out several mistakes and valuable comments.
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