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.
- Floer A, Weinstein A: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 1986, 69: 397-408. 10.1016/0022-1236(86)90096-0MathSciNetView ArticleMATHGoogle Scholar
- Oh Y-G:Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class . Commun. Partial Differ. Equ. 1988, 13: 1499-1519. 10.1080/03605308808820585View ArticleMATHGoogle Scholar
- Oh Y-G:Corrections to ‘Existence of semi-classical bound state of nonlinear Schrödinger equations with potentials of the class ’. Commun. Partial Differ. Equ. 1989, 14: 833-834.MATHGoogle Scholar
- Ambrosetti A, Badiale M, Cingolani S: Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 1997, 140: 285-300. 10.1007/s002050050067MathSciNetView ArticleMATHGoogle Scholar
- Del Pino M, Felmer P: Local mountain passes for semilinear elliptic problems in unbounded domains. Calc. Var. 1996, 4: 121-137. 10.1007/BF01189950MathSciNetView ArticleMATHGoogle Scholar
- Del Pino M, Felmer P: Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting. Indiana Univ. Math. J. 1999, 48(3):883-898.MathSciNetView ArticleMATHGoogle Scholar
- Rabinowitz P: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 1992, 43: 270-291. 10.1007/BF00946631MathSciNetView ArticleMATHGoogle Scholar
- Wang X: On concentration of positive bound states of nonlinear Schrödinger equations. Commun. Math. Phys. 1993, 153: 229-244. 10.1007/BF02096642View ArticleMATHGoogle Scholar
- Wang X, Zeng B: On concentration of positive bound states of nonlinear Schrödinger equation with competing potential functions. SIAM J. Math. Anal. 1997, 28: 633-655. 10.1137/S0036141095290240MathSciNetView ArticleMATHGoogle Scholar
- Del Pino M, Felmer P: Multi-peak bound states of nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1998, 15(2):127-149. 10.1016/S0294-1449(97)89296-7MathSciNetView ArticleMATHGoogle Scholar
- Gui C: Existence of multi-bump solutions for nonlinear Schrödinger equations. Commun. Partial Differ. Equ. 1996, 21: 787-820. 10.1080/03605309608821208MathSciNetView ArticleMATHGoogle Scholar
- Lin C, Ni W-M, Takagi I: Large amplitude stationary solutions to a chemotaxis systems. J. Differ. Equ. 1988, 72: 1-27. 10.1016/0022-0396(88)90147-7MathSciNetView ArticleMATHGoogle Scholar
- Ni WM, Takagi I: On the shape of least-energy solutions to a semilinear Neumann problem. Commun. Pure Appl. Math. 1990, 45: 819-851.MathSciNetMATHGoogle Scholar
- Ni WM, Takagi I: Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math. J. 1993, 70: 247-281. 10.1215/S0012-7094-93-07004-4MathSciNetView ArticleMATHGoogle Scholar
- Wang ZQ: On the existence of multiple, single peaked solutions for a semilinear Neumann problem. Arch. Ration. Mech. Anal. 1992, 120: 375-399. 10.1007/BF00380322View ArticleMATHMathSciNetGoogle Scholar
- Gui C: Multi-peak solutions for a semilinear Neumann problem. Duke Math. J. 1996, 84: 739-769. 10.1215/S0012-7094-96-08423-9MathSciNetView ArticleMATHGoogle Scholar
- Li YY: On a singularly perturbed equation with Neumann boundary condition. Commun. Partial Differ. Equ. 1998, 23: 487-545.MATHMathSciNetGoogle Scholar
- Wei J, Winter M: Multiple boundary spike solutions for a wide class of singular perturbation problems. J. Lond. Math. Soc. 1999, 59(2):585-606. 10.1112/S002461079900719XMathSciNetView ArticleGoogle Scholar
- Cingolani S, Lazzo M: Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J. Differ. Equ. 2000, 160: 118-138. 10.1006/jdeq.1999.3662MathSciNetView ArticleMATHGoogle Scholar
- Ambrosetti A, Malchiodi A: Nonlinear Analysis and Semilinear Elliptic Problems. Cambridge University Press, Cambridge; 2007.View ArticleMATHGoogle Scholar
- Liu, W, Zhao, P: Critical semilinear Neumann problem with magnetic fields. PreprintGoogle Scholar
- Berestycki H, Lions PL: Nonlinear scalar field equations I. Existence of a ground state. Arch. Ration. Mech. Anal. 1983, 82: 313-345.MathSciNetMATHGoogle Scholar
- Zhang J, Zou W: A Berestycki-Lions theorem revisited. Commun. Contemp. Math. 2012., 14(5): Article ID 1250033MATHGoogle Scholar
- Lions PL: The concentration-compactness principle in the calculus of variation. The locally compact case. II. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1984, 1: 223-283.MATHGoogle Scholar
- Alves CO, Figueiredo GM, Furtado MF: Multiple solutions for a nonlinear Schrödinger equation with magnetic fields. Commun. Partial Differ. Equ. 2011, 36: 1565-1586. 10.1080/03605302.2011.593013MathSciNetView ArticleMATHGoogle Scholar
- Brezis H, Lieb E: A relation between pointwise convergence of functions and convergence of functional. Proc. Am. Math. Soc. 1983, 88: 486-490.MathSciNetView ArticleMATHGoogle Scholar
- Lions PL: The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoam. 1985, 1: 145-201.View ArticleMATHMathSciNetGoogle Scholar
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.