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

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, $0<\gamma \le 1$ 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.

Cited by in corpus (1)

• Anzellotti's pairing theory and the Gauss--Green theorem