W 0 1,p ( Ω ) × W 0 1,p ( Ω ) versus C 0 1 ( Ω ) × C 0 1 ( Ω ) local minimizers

Abstract

In this work, we consider a functional $\mathit{I}:{\mathit{W}}_0^{1\mathrm{,}\mathit{p}}\mathrm{(}\mathrm{\Omega }\mathrm{)}\times {\mathit{W}}_0^{1\mathrm{,}\mathit{p}}\mathrm{(}\mathrm{\Omega }\mathrm{)}\to \mathbb{R}$ of the form $$\begin{array}{c}\mathit{I}\mathrm{(}\mathit{u}\mathrm{,}\mathit{v}\mathrm{)}={\displaystyle \frac{1}{\mathit{p}}}{{\displaystyle \int }}_{\mathrm{\Omega }}\mathrm{(}|\nabla \mathit{u}{|}^{\mathit{p}}+|\nabla \mathit{v}{|}^{\mathit{p}}\mathrm{)}\mathit{d}\mathit{x}-{{\displaystyle \int }}_{\mathrm{\Omega }}\mathit{H}\mathrm{(}\mathit{x}\mathrm{,}\mathit{u}\mathrm{(}\mathit{x}\mathrm{)}\mathrm{,}\mathit{v}\mathrm{(}\mathit{x}\mathrm{)}\mathrm{)}\mathit{d}\mathit{x}\end{array}$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^{\mathit{N}}$ is a smooth bounded domain, $max\{|{\partial }_{\mathit{s}}\mathit{H}\mathrm{(}\mathit{x}\mathrm{,}\mathit{s}\mathrm{,}\mathit{t}\mathrm{)}|\mathrm{,}|{\partial }_{\mathit{t}}\mathit{H}\mathrm{(}\mathit{x}\mathrm{,}\mathit{s}\mathrm{,}\mathit{t}\mathrm{)}|\}⩽\mathit{C}\mathrm{(}1+|\mathit{s}{|}^{{\mathit{\sigma }}_1-1}+|\mathit{t}{|}^{{\mathit{\sigma }}_2-1}\mathrm{)}$ a.e. $\mathit{x}\in \mathrm{\Omega }$ , for some $\mathit{C}\mathrm{>}0$ , $\forall \mathit{t}\mathrm{,}\mathit{s}\in \mathbb{R}$ , $\mathit{p}\mathrm{<}{\mathit{\sigma }}_{\mathit{i}}⩽{\mathit{p}}^{\ast }:=\mathit{N}\mathit{p}\mathrm{/}\mathrm{(}\mathit{N}-\mathit{p}\mathrm{)}$ , $\mathit{i}=1\mathrm{,}2$ , and $1\mathrm{<}\mathit{p}\mathrm{<}\mathit{N}$ . We prove that a local minimum in the topology of ${\mathit{C}}_0^1\mathrm{(}\mathrm{\Omega }\mathrm{)}\times {\mathit{C}}_0^1\mathrm{(}\mathrm{\Omega }\mathrm{)}$ is a local minimum in the topology of ${\mathit{W}}_0^{1\mathrm{,}\mathit{p}}\mathrm{(}\mathrm{\Omega }\mathrm{)}\times {\mathit{W}}_0^{1\mathrm{,}\mathit{p}}\mathrm{(}\mathrm{\Omega }\mathrm{)}$ . An important application of this result is related to the question of multiplicity of solutions for a class of systems with concave-convex type nonlinearities.