Anisotropic p, q-laplacian equations when p goes to 1

Abstract

In this paper we prove a stability result for an anisotropic elliptic problem. More precisely, we consider the Dirichlet problem for an anisotropic equation, which is as the p–Laplacian equation with respect to a group of variables and as the q–Laplacian equation with respect to the other variables (1 < p < q), with datum f belonging to a suitable Lebesgue space. 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, regardless of the size of the datum f. Moreover, we prove that this u is the unique solution of a limit problem having the 1–Laplacian operator with respect to the first group of variables. Furthermore, the regularity of the solutions to the limit problem is studied and explicit examples are shown.