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PERTURBATIONS FROM INDEFINITE SYMMETRIC ELLIPTIC BOUNDARY VALUE PROBLEMS

Part of: Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  08 February 2017

LIANG ZHANG  and
XIANHUA TANG
LIANG ZHANG
Affiliation:
School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P.R. China e-mail: [email protected]
XIANHUA TANG
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, P.R. China
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Abstract

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In this paper, we study the multiplicity of solutions for the following problem:

$$\begin{equation*} \begin{cases} -\Delta u-\Delta(|u|^{\alpha})|u|^{\alpha-2}u=g(x,u)+\theta h(x,u), \ \ x\in \Omega,\\ u=0, \ \ x\in \partial\Omega, \end{cases} \end{equation*}$$
where α ≥ 2, Ω is a smooth bounded domain in ${\mathbb{R}}$N, θ is a parameter and g, h ∈ C($\bar{\Omega}$ × ${\mathbb{R}}$). Under the assumptions that g(x, u) is odd and locally superlinear at infinity in u, we prove that for any j ∈ $\mathbb{N}$ there exists ϵj > 0 such that if |θ| ≤ ϵj, the above problem possesses at least j distinct solutions. Our results generalize some known results in the literature and are new even in the symmetric situation.

MSC classification

Primary: 35B20: Perturbations
Secondary: 35J20: Variational methods for second-order elliptic equations 35J62: Quasilinear elliptic equations
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 3 , September 2017 , pp. 635 - 648
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

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