Infinitely Many Solutions for a Class of Quasi-linear Elliptic Problem

IF 0.9 4区 数学 Q3 MATHEMATICS, APPLIED
Xiao-yao Jia, Zhen-luo Lou
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引用次数: 0

Abstract

In this paper, we study the following quasi-linear elliptic equation

$$\left\{{\matrix{{- \,{\rm{div(}}\phi {\rm{(}}\left| {\nabla u} \right|{\rm{)}}\nabla u{\rm{) = \lambda}}\psi {\rm{(}}\left| u \right|{\rm{)}}u + \,\varphi {\rm{(}}\left| u \right|{\rm{)}}u,\,\,\,\,{\rm{in}}\,\,\,\Omega,\,\,\,} \cr {u = 0,\,\,\,\,\,\,\,{\rm{on}}\,\,\partial \Omega {\rm{,}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \cr}} \right.$$

where Ω ⊂ ℝN is a bounded domain, λ > 0 is a parameter. The function ψ(∣t∣)t is the subcritical term, and ϕ(∣t∣)t is the critical Orlicz-Sobolev growth term with respect to φ. Under appropriate conditions on φ, ψ and ϕ, we prove the existence of infinitely many weak solutions for quasi-linear elliptic equation, for λ ∈ (0, λ0), where λ0 > 0 is a fixed constant.

一类准线性椭圆问题的无限多解
本文研究了以下准线性椭圆方程、{\rm{div(}}\phi {\rm{(}}\left| {\nabla u} \right|\rm{)}}\nabla u{rm{) = \lambda}}\psi {\rm{(}}\left| u \right|\rm{)}}u + \,\varphi {\rm{(}}\left| u \right|\rm{)}}u、\cr {u = 0,\,\,\,{/rm{on}}\,\,\partial\Omega{/rm{,}}\,\,\,\,\、\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \cr}}\其中 Ω ⊂ ℝN 是一个有界域,λ > 0 是一个参数。函数 ψ(∣t∣)t 是次临界项,ϕ(∣t∣)t 是关于 φ 的临界 Orlicz-Sobolev 增长项。在φ、ψ和ϕ的适当条件下,我们证明了准线性椭圆方程在λ∈ (0, λ0)(其中λ0 >0是一个固定常数)时存在无穷多个弱解。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
70
审稿时长
3.0 months
期刊介绍: Acta Mathematicae Applicatae Sinica (English Series) is a quarterly journal established by the Chinese Mathematical Society. The journal publishes high quality research papers from all branches of applied mathematics, and particularly welcomes those from partial differential equations, computational mathematics, applied probability, mathematical finance, statistics, dynamical systems, optimization and management science.
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