Existence of solutions to a 1-Laplacian problem with a concave-convex nonlinearity

Abstract

In this paper, we analyze a “concave-convex” type problem involving the 1-Laplacian operator in a general Lipschitz–continuous domain and prove the existence of two positive solutions. Owing to 1-Laplacian is 0-homogeneous, the “concave” term must be singular. Hence, we should deal with an energy functional having two non–differentiable terms: the total variation and that one coming from the singular term. Due to these difficulties, we do not get solutions as critical points of the energy functional defined in the BV(Ω) space. Instead, we study problems involving the p-Laplacian operator and let p go to 1.