Elliptic problems involving the 1--Laplacian and a singular lower order term
De Cicco, Giachetti, Segura de Leon · arXiv (Cornell University) · 2017
This paper is concerned with the Dirichlet problem for an equation involving the 1--Laplacian operator $Δ_1 u$ and having a singular term of the type $\frac{f(x)}{u^γ}$. Here $f\in L^N(Ω)$ is nonnegative, $00$ 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.