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

created by decicco on 15 May 2017
modified on 21 Aug 2018

[BibTeX]

Accepted Paper

Inserted: 15 may 2017
Last Updated: 21 aug 2018

Journal: J. of London Math. Soc.
Year: 2018

Abstract:

This paper is concerned with the Dirichlet problem for an equation involving the $1$-Laplacian operator $\Delta_1 u:={\rm{div}}\left(\frac{Du}{ Du }\right)$ 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.