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Open Access Research

Least energy solutions for a quasilinear Schrödinger equation with potential well

Yujuan Jiao

Author affiliations

College of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou, 730124, P.R. China

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, P.R. China

Citation and License

Boundary Value Problems 2013, 2013:9  doi:10.1186/1687-2770-2013-9

Published: 21 January 2013

Abstract

In this paper, we consider the existence of least energy solutions for the following quasilinear Schrödinger equation:

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M2">View MathML</a> having a potential well, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M4">View MathML</a> is a parameter. Under suitable hypotheses, we obtain the existence of a least energy solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M5">View MathML</a> of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M6">View MathML</a>) which localizes near the potential well <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M7">View MathML</a> for λ large enough by using the variational method and the concentration compactness method in an Orlicz space.

MSC: 35J60, 35B33.

Keywords:
quasilinear Schrödinger equation; least energy solution; Orlicz space; concentration compactness method; variational method