Elliptic equations involving the $1$-Laplacian and a subcritical source term

Abstract

In this paper we deal with a Dirichlet problem for an elliptic equation involving the $1$-Laplacian operator and a source term. We prove that, when the growth of the source is subcritical, there exist two bounded nontrivial solutions to our problem. Moreover, a Pohozaev type identity is proved, which holds even when the growth is supercritical. We also show explicit examples of our results.

Nonlinear elliptic equations, 1-Laplacian operator, subcritical source term, p-Laplacian operator, mountain pass geometry