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

TL;DR

This paper investigates elliptic equations with the 1-Laplacian operator and subcritical source terms, establishing existence of solutions and a Pohozaev identity applicable even in supercritical growth scenarios.

Contribution

It proves the existence of two solutions for subcritical sources and extends Pohozaev identities to supercritical growth cases.

Findings

Existence of two bounded solutions for subcritical sources

Pohozaev identity valid for supercritical growth

Explicit examples illustrating the results

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.