Kirchhoff type elliptic equations with double criticality in Musielak-Sobolev spaces

Abstract

This paper aims to establish the existence of a weak solution for the non-local problem:where Ω ⊆ R N , N ≥ 2 is a bounded and smooth domain containing two open and connected subsets Ωp and Ω N such that Ωp ∩ Ω N = ∅ and ∆ H u = div(h(x, |∇u|)∇u) is the H-Laplace operator. We assume that ∆ H reduces to ∆ p(x) in Ωp and to ∆ N in Ω N , the non-linear function f : Ω × R → R act as |t| p * (x)−2 t on Ωp and as e α|t| N/(N −1) on Ω N for sufficiently large |t|. To establish our existence results in a Musielak-Sobolev space, we use a variational technique based on the mountain pass theorem.