由p-Laplacian算子导出的半变分不等式的特征值问题

Eigenvalue Problems for Hemivariational Inequality Driven by the p-Laplacian

研究了一类具有非光滑局部Lipschitz位势(半变分不等式)的非线性特征值问题其中1<p<∞,Ω R^N是有界区域.目的是把最近的超线性(即p=2)问题的非平凡解存在性和连续性结果推广到一般情况(即1<p<∞).不仅推广了Miyagaki and Souto研究工作[Superlinear Problems without Ambrosetti and Rabinowitz GrowthCondition,Jour.Diff.Equa.,245(2008)3628-3638],同时也推广了Schechter和Zou的研究工作[Superlinear Problems,Pacific J.Math.,214(2004)145-160].本文使用的方法基于局部Lipschitz函数的非光滑临界点理论.

We consider a kind of nonlinear eigenvalue problem driven by the p-laplacian with a nonsmooth locally Lipschitz potential （hemivariational inequality）, that is, {-div（｜｜△x（z）｜｜R^N^p-2△x（z））∈λδj（z,x（z））,z∈Ω x（z）=0,z∈δΩ,where 1〈p〈∞,Ω R^n is a bounded domain. The purpose of this paper is to extend earlier existence and continuation results of nontrivial solutions of the problem in the superline case （i.e., p = 2） to the general case （i.e., 1 〈 p 〈 ∞）. We not only extend the existence results of nontrivial solutions for almost every parameter A due to Miyagaki and Souto [Superlinear Problems without Ambrosetti and Rabinowitz Growth Condition, Jour. Diff. Equa., 245 （2008） 3628-3638], but also extend the existence results of nontrivial solutions for every parameter λ due to Schechter and Zou [Superlincar Problems, Pacific Y. Math., 214 （2004） 145-160] to the general case when 1 〈 p 〈 ∞. Our approach is based on the non-smooth critical point theory for locally Lipschitz functions.