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Multiple Solutions of Sublinear Quasilinear Schrödinger Equations with Small Perturbations

Part of: Equations of mathematical physics and other areas of application Elliptic equations and systems

Published online by Cambridge University Press:  29 November 2018

Liang Zhang ,
X. H. Tang  and
Yi Chen
Liang Zhang*
Affiliation:
School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, People's Republic of China ([email protected])
X. H. Tang
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, People's Republic of China ([email protected])
Yi Chen
Affiliation:
Department of Mathematics, China University of Mining and Technology, Xuzhou 221116, People's Republic of China ([email protected])
*
*Corresponding author.
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Abstract

In this paper, we consider the existence of multiple solutions for the quasilinear Schrödinger equation

$$\left\{ {\matrix{ {-\Delta u-\Delta (\vert u \vert ^\alpha )\vert u \vert ^{\alpha -2}u = g(x,u) + \theta h(x,u),\;\;x\in \Omega } \hfill \cr {u = 0,\;\;x\in \partial \Omega ,} \hfill \cr } } \right.$$
where Ω is a bounded smooth domain in ℝN (N ≥ 1), α ≥ 2 and θ is a parameter. Under the assumption that g(x, u) is sublinear near the origin with respect to u, we study the effect of the perturbation term h(x, u), which may break the symmetry of the associated energy functional. With the aid of critical point theory and the truncation method, we show that this system possesses multiple small negative energy solutions.

Keywords

quasilinear Schrödinger equationsmultiple solutionscritical point theory

MSC classification

Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 35J65: Nonlinear boundary value problems for linear elliptic equations 35Q55: NLS-like equations (nonlinear Schrödinger)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 2 , May 2019 , pp. 471 - 488
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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