Barrier solutions of elliptic variational inequalities

Abstract

We consider elliptic variational inequalities in a bounded domain Ω⊂RN of the formu∈K:〈Au+λ|u|p−2u+F(u),v−u〉≥〈h,v−u〉,∀v∈K, where A is a second order quasilinear elliptic operator of divergence type, F is the operator generated by lower order terms, and K is a closed convex subset of the Sobolev space W1,p(Ω), 1<p<∞, and λ≥0. The main goal of this paper is to answer the following question: Does the variational inequality possess barrier solutions? Here, solutions u∗ and u∗ of the variational inequality are called barrier solutions if u∗≤u∗, and any solution u of the variational inequality satisfies u∗≤u≤u∗. In other words, we are going to provide sufficient conditions which ensure that the solution set S of the variational inequality is nonempty, and S possesses a greatest and a smallest element with respect to the natural partial ordering of functions. An answer to the raised question is by no means trivial as will be seen by specific examples of the variational inequality under consideration. The obtained results are finally used to derive a priori order bounds as well as existence of barrier solutions for multi-valued variational inequalities including variational–hemivariational inequalities as special case.