On the existence of nontrivial solutions of inequalities in Orlicz-Sobolev spaces

Abstract

This paper is about an alternate variational inequality formulation for the boundary value problem$$\begin{array}{l}-{\rm div} (a(|\nabla u|) \nabla u) + \partial_u G(x,u) \ni 0 \;\mbox{ in } \;\Omega , \\u=0 \;\mbox{ on } \;\partial\Omega ,\end{array}$$where the principal part may have non-polynomial or very slow growth. As a consequence of this formulation, we can apply abstract nonsmooth linking theorems to study the existence and multiplicity of nontrivial solutions to the above problem.