arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 0912.3465math.AP
收藏

具有临界增长的$p(x)-$ Laplace算子的多解性

Multiple solutions for the $p(x)-$laplace operator with critical growth

Analía Silva

阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

本文的目的是将以前关于临界增长的拟线性椭圆型问题解的多重性的结果推广到变指数情形。在\cite{DPFBS}的思想下，证明了在$\partialΩ$上具有齐次Dirichlet边界条件的拟线性椭圆型方程$-Δ_{p(x)} u = |u|^{q(x)-2}u +λf(x,u)$在$\R^N$的光滑有界区域$Ω$中至少存在三个非平凡解.我们假设$\{q(x)=p^*(x)\}\not=\emptyset$，其中$p^*(x)=Np(x)/(N-p(x))$是可变指数的临界Sobolev指数，$Δ_{p(x)} u = {div}(|\nabla u|^{p(x)-2}\nabla u)$是$p(x)-$拉普拉斯算子。证明是基于变分变元和变指数空间的浓度紧性方法的推广。

英文摘要：

The aim of this paper is to extend previous results regarding the multiplicity of solutions for quasilinear elliptic problems with critical growth to the variable exponent case. We prove, in the spirit of \cite{DPFBS}, the existence of at least three nontrivial solutions to the following quasilinear elliptic equation $-Δ_{p(x)} u = |u|^{q(x)-2}u +λf(x,u)$ in a smooth bounded domain $Ω$ of $\R^N$ with homogeneous Dirichlet boundary conditions on $\partialΩ$. We assume that $\{q(x)=p^*(x)\}\not=\emptyset$, where $p^*(x)=Np(x)/(N-p(x))$ is the critical Sobolev exponent for variable exponents and $Δ_{p(x)} u = {div}(|\nabla u|^{p(x)-2}\nabla u)$ is the $p(x)-$laplacian. The proof is based on variational arguments and the extension of concentration compactness method for variable exponent spaces.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑