Nodal solutions to quasilinear elliptic problems involving the 1-Laplacian operator via variational and approximation methods

Abstract

In this work we use two different methods to get nodal solutions to quasilinear elliptic problems involving the 1−Laplacian operator. In the first one, we develop an approach based on a minimization of the energy functional associated to a problem involving the 1−Laplacian operator in R , on a subset of the Nehari set which contains just sign-changing functions. In the second part we obtain a nodal solution to a quasilinear elliptic problem involving the 1−Laplacian operator in a bounded domain, through a thorough analysis of the sequence of solutions of the p−Laplacian problem associated to it, as p → 1. In both cases, several technical difficulties appear in comparison with the related results involving signed solutions.