High perturbations of quasilinear problems with double criticality

This paper is concerned with the qualitative analysis of solutions to the following class of quasilinear problems {-ΔΦu=f(x,u)inΩ,u=0on∂Ω,where ΔΦu=div(φ(x,|∇u|)∇u) and Φ(x,t)=∫0|t|φ(x,s)sds is a generalized N-function. We assume that Ω ⊂ R^N is a smooth bounded domain that contains two open regions Ω _N, Ω _p with Ω ¯ _N∩ Ω ¯ _p= ∅. The features of this paper are that - Δ _Φu behaves like - Δ _Nu on Ω _N and - Δ _pu on Ω _p, and that the growth of f: Ω × R→ R is like that of eα|t|NN-1 on Ω _N and as |t|p∗-2t on Ω _p when |t| is large enough. The main result establishes the existence of solutions in a suitable Musielak–Sobolev space in the case of high perturbations with respect to the values of a positive parameter.