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arXiv · 1711.06812

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

De Cicco·Giachetti·Segura de Leon
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Abstract

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, $0<γ\le1$ and $Ω$ 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.

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De Cicco, Giachetti, Segura de Leon. 2017-11-18. Elliptic problems involving the 1--Laplacian and a singular lower order term. https://arxiv.org/abs/1711.06812

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