Multiplicity of Solutions to Elliptic Problems Involving the 1-Laplacian with a Critical Gradient Term

Abstract

In the present paper we study the Dirichlet problem for an equation involving the -Laplacian and a total variation term as reaction. We prove a strong multiplicity result. Namely, we show that for any positive Radon measure concentrated in a set away from the boundary and singular with respect to a certain capacity, there exists an unbounded solution, and measures supported on disjoint sets generate different solutions. These results can be viewed as the analogue for the -Laplacian operator of some known multiplicity results which were first obtained by Ireneo Peral, to whom this article is dedicated, and his collaborators.

Elliptic equations involving the 1-Laplacian and a total variation term with $L^{N,\infty}$-data

“…We point out that the situation concerning existence is rather similar to that shown in studying problem (2)    −∆u + |∇u| 2 = λ u |x| 2 in Ω , u = 0 on ∂Ω , in domains satisfying 0 ∈ Ω, since the presence of the quadratic gradient term induces a regularizing effect (see [3] and [1], see also Remark 5.4 below). Indeed, existence of a positive solution to (2) can be proved for all λ > 0, while if the gradient term does not appear, solutions can be expected only for λ small enough, due to Hardy's inequality.…”

1-Laplacian type problems with strongly singular nonlinearities and gradient terms

“…Remark 3.6. As will be clear by the proof of Lemma 3.12 below, the fact that solutions of problem (3.1) do not possess jump parts is a regularizing effect given by the presence of the gradient term g(u)|Du|; a similar situation was noticed in [42,1,40] while in the case g(s) ≡ 0 solutions can have a nontrivial jump part (see [23,24]). Also the fact that u is bounded is quite natural and this is essentially due to the presence of the zero order term h(u) f .…”

Regularity of renormalized solutions to nonlinear elliptic equations away from the support of measure data

“…Our motivation to study this kind of problems comes from searching nonregular solutions to equations with a gradient term having "natural" growth which, by means of the Cole-Hopf change of unknown, are reduced to (1.1) (see Abdellaoui, Dall'Aglio and Peral [2] and Abdel Hamid and Bidaut-Veron [1]). In fact, we apply the results of this paper to obtain nonregular solutions to an equation involving the 1-Laplacian and a total variation term in Abdellaoui, Dall'Aglio and Segura de León, see [3]. Two features of problem (1.1) deserve a comment.…”