On a Φ-Kirchhoff multivalued problem with critical growth in an Orlicz–Sobolev space

This paper is concerned with the multiplicity of nontrivial solutions in an Orlicz–Sobolev space for a nonlocal problem with critical growth, involving N-functions and theory of locally Lipschitz continuous functionals. More precisely, in this paper, we study a result of multiplicity to the following multivalued elliptic problem:

−M(∫_ΩΦ(|∇u|) dx)Δ_Φu∈∂F(·,u)+αh(u)   in Ω,

u∈W₀¹L_Φ(Ω),

where Ω⊂R^N is a bounded smooth domain, N≥3, M is a continuous function, Φ is an N-function, h is an odd increasing homeomorphism from R to R, α is positive parameter, Δ_Φ is the corresponding Φ-Laplacian and ∂F(·,t) stands for Clarke generalized of a function F linked with critical growth. We use genus theory to obtain the main result.