Solutions for nonlinear variational inequalities with a nonsmooth potential

Michael E Filippakis,Nikolaos S Papageorgiou

doi:10.1155/s1085337504312017

Michael E Filippakis, Nikolaos S Papageorgiou

Open Access

https://doi.org/10.1155/s1085337504312017

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Abstract

First we examine a resonant variational inequality driven by the p‐Laplacian and with a nonsmooth potential. We prove the existence of a nontrivial solution. Then we use this existence theorem to obtain nontrivial positive solutions for a class of resonant elliptic equations involving the p‐Laplacian and a nonsmooth potential. Our approach is variational based on the nonsmooth critical point theory for functionals of the form φ = φ1 + φ2 with φ1 locally Lipschitz and φ2 proper, convex, lower semicontinuous.

Highlights

In this paper, we consider the following nonlinear variational inequality at resonance with a nonsmooth potential function (Z ⊆ RN is a bounded domain with a C2-boundary ∂Z):Z Dx(z) p−2 Dx(z), D y(z) − Dx(z) RN dz − λ1 Z x(z) p−2x(z)(y − x)(z)dz ≥ u(z)(y − x)(z)dz ∀y ∈ C, Z (1.1)where C = {x ∈ W01,p(Z) : x(z) ≥ g(z) a.e on Z}, with g ∈ W1,p(Z), g(z) ≤ 0 a.e. on Z, and u ∈ Lq(Z), (1/ p + 1/q = 1, 1 < p < ∞), u(z) ∈ ∂ j(z, x(z)) a.e. on Z
First we examine a resonant variational inequality driven by the p-Laplacian and with a nonsmooth potential
We consider the following nonlinear variational inequality at resonance with a nonsmooth potential function (Z ⊆ RN is a bounded domain with a C2-boundary ∂Z): Z Dx(z) p−2 Dx(z), D y(z) − Dx(z) RN dz − λ1 Z x(z) p−2x(z)(y − x)(z)dz ≥ u(z)(y − x)(z)dz ∀y ∈ C, Z

Summary

Introduction

We consider the following nonlinear variational inequality at resonance with a nonsmooth potential function (Z ⊆ RN is a bounded domain with a C2-boundary ∂Z):. We study the following nonlinear elliptic problem at resonance with nonsmooth potential:. In [19], the problem under consideration is semilinear (i.e., p = 2), not in resonance, g ≡ 0, and the potential function is smooth (i.e., a C1 function) The right-hand side nonlinearity is Caratheodory ( the corresponding potential function is C1) and it may depend on the gradient of the unknown function (see Le [13]) His approach is based on the method of upper and lower solutions. We refer to the books of Clarke [4] and Denkowski et al [6]

Mathematical background

Generalized variational inequalities

Positive solutions