Local $T$-sets and renormalized solutions of degenerate quasilinear elliptic equations with an $L^1$-datum

Abstract

In this paper we study essentially the questions of uniqueness and stability of solutions of boundary value problems associated with equations of the type: $$ \text{div}(\hat a(x,u,\nabla u))+b(x)|u|^{\gamma-1}u=\mu\in L^1(\Omega) $$ on an arbitrary open subset $\Omega$ of $\mathbb{R}^N$ with $\hat a(x,u,\nabla u)$ a Carathéodory nonlinear function satisfying the general conditions of Leray-Lions where the coerciveness condition is weakened to allow degeneracies and becomes $$ \hat a(x,u,\xi)\cdot\xi\ge a(x)|\xi|^p,\ \forall u\in\mathbb{R},\ \forall\xi\in\mathbb{R}^N, \ \mbox{and $x$ a.e. in } \Omega, $$ with $p>1$ an arbitrary real number and $a$ an $L^1$-weight which might vanish or go to infinity on $S$, a closed subset of $\overline\Omega$ whose measure is zero. Here $b$ is an $L^1$-nonnegative function with properties similar to those of the weight $a$, $\gamma$ a positive number belonging to suitable intervals. For $p>1$ arbitrary and general weight $a$, we need new functional sets called "local T-sets" which are extensions of local Sobolev spaces. The "localization" is to handle the degeneracy. We get uniqueness and stability for $S$ satisfying a geometrical condition or $S$ and $a$ an analytic-geometrical one.