On the behaviour of solutions to the Dirichlet problem for the $p(x)$-Laplacian when $p(x)$ goes to $1$ in a subdomain

Abstract

In this paper we prove a stability result for some classes of elliptic problems involving variable exponents. More precisely, we consider the Dirichlet problem for an elliptic equation in a domain $\Omega$, which is the $p$--Laplacian equation, $-\mbox{div}(|\nabla u|^{p-2} \nabla u) =f$, in a subdomain $\Omega_1$ of $\Omega$ and the Laplace equation, $-\Delta u = f$, in its complementary (that is, our equation involves the so-called $p(x)$--Laplacian with a discontinuous exponent). We assume that the right-hand side $f$ belongs to $L^\infty(\Omega)$. For this problem, we study the behaviour of the solutions as $p$ goes to $1$, showing that they converge to a function $u$, which is almost everywhere finite when the size of the datum $f$ is small enough. Moreover, we prove that this $u$ is a solution of a limit problem involving the $1$-Laplacian operator in $\Omega_1$. We also discuss uniqueness under a favorable geometry.