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 problem { − Δ 1 u + ξ u | u | = λ | u | q − 2 u + | u | 1 ⁎ − 2 u , in  Ω , u = 0 , on  ∂ Ω , where 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.