Multiplicity of solutions for a class of quasilinear problems involving the $1$-Laplacian operator with critical growth

Abstract

The aim of this paper is to establish two results about multiplicity of solutions to problems involving the 1−Laplacian operator, with nonlinearities with critical growth. To be more specific, we study the following problemwhere Ω is a smooth bounded domain in R N , N ≥ 2 and ξ ∈ {0, 1}. Moreover, λ > 0, q ∈ (1, 1 * ) and 1 * = N N −1 . The first main result establishes the existence of many rotationally non-equivalent and nonradial solutions by assuming that ξ = 1, Ω = {x ∈ R N : r < |x| < r + 1}, N ≥ 2, N = 3 and r > 0. In the second one, Ω is a smooth bounded domain, ξ = 0, and the multiplicity of solutions is proved through an abstract result which involves genus theory for functionals which are sum of a C 1 functional with a convex lower semicontinuous functional.2010 Mathematics Subject Classification. 1-Laplacian operator; Functions of bounded variation, Varitional Methods.