Elliptic problems involving the 1--Laplacian and a singular lower order term

TL;DR

This paper establishes existence and uniqueness results for solutions to a singular elliptic PDE involving the 1-Laplacian operator, using limit processes from p-Laplacian problems and extending divergence-measure theory.

Contribution

It introduces a new concept of solution for the 1-Laplacian with singular terms and proves existence, uniqueness under certain conditions, and provides explicit examples showing non-uniqueness.

Findings

Existence of solutions via limit of p-Laplacian problems.

Uniqueness when f > 0 a.e.

Counterexamples demonstrating non-uniqueness.

Abstract

This paper is concerned with the Dirichlet problem for an equation involving the 1--Laplacian operator $\Delta_1 u$ and having a singular term of the type $\frac{f(x)}{u^\gamma}$. Here $f\in L^N(\Omega)$ is nonnegative, $0<\gamma\le1$ and $\Omega$ is a bounded domain with Lipschitz--continuous boundary. We prove an existence result for a concept of solution conveniently defined. The solution is obtained as limit of solutions of $p$--Laplacian type problems. Moreover, when $f(x)>0$ a.e., the solution satisfies those features that might be expected as well as a uniqueness result. We also give explicit 1--dimensional examples that show that, in general, uniqueness does not hold. We remark that the Anzellotti theory of $L^\infty$--divergence--measure vector fields must be extended to deal with this equation.