Multiplicity of solutions to elliptic problems involving the 1-Laplacian with a critical gradient term

In the present paper we study the Dirichlet problem for an equation involving the 1–Laplacian and a total variation term as reaction. We prove a strong multiplicity result: 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 1–Laplacian operator of some known multiplicity results which were first obtained by Ireneo Peral, to whom this article is dedicated, and his collaborators.