H1∩Lp versus C1 Local Minimizers

Abstract

We show that a local minimizer of $\mathrm{\Phi }$ in the $C^1$ topology must be a local minimizer in the $H^1\cap L^p$ topology, under suitable assumptions for the functional $\Phi =(1/2){{\displaystyle {\int }_{\Omega }|\nabla u|}}^2+(1/p){{\displaystyle {\int }_{\Omega }|u|}}^p-{\displaystyle {\int }_{\Omega }F(x,u)}$ with supercritical exponent $p>2^{\ast }=2n/(n-2)$. This result can be used to establish a solution to the corresponding equation admitting sub- and supersolution. Hence, we extend the conclusion proved by Brezis and Nirenberg (1993), the subcritical and critical case.