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 1-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 1-Laplacian operator of some known multiplicity results which were first obtained by Ireneo Peral, to whom this article is dedicated, and his collaborators.

Keywords

• nonlinear elliptic problems
• nonuniqueness
• problems with total variation
• cole–hopf change
• 35j15
• 35j75
• 35j60
• 35a02