Some existence results and properties of solutions in quasilinear variational inequalities with general growths

Abstract

This paper is about the existence and some properties of solutions of the boundary value problem: $$ \left\{ \begin{gathered} - div(a(|\nabla u|)\nabla u) = f(x,u)in\Omega \hfill \\ u = 0on\partial \Omega , \hfill \\ \end{gathered} \right. $$ with the principal term a(|∇u|)∇u having general growth. In the non-coercive case, a sub-supersolution approach is applied to get existence and enclosure results. Other properties such as compactness of solution sets and existence of extremal solutions are also derived.