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Open Access Research

Multiple solutions to nonlinear Schrödinger equations with critical growth

Wulong Liu1* and Peihao Zhao2

Author Affiliations

1 Department of Mathematics, Jiangxi University of Science and Technology, Ganzhou, 341000, P.R. China

2 School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, P.R. China

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Boundary Value Problems 2013, 2013:199  doi:10.1186/1687-2770-2013-199

Published: 4 September 2013

Abstract

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M1">View MathML</a>, which is a key condition in their paper. In the proofs we apply variational methods and Ljusternik-Schnirelmann theory.

MSC: 35J60, 35Q55.

Keywords:
nonlinear Schrödinger equations; critical growth; variational methods