On a class of quasilinear elliptic problems involving Trudinger-Moser nonlinearities

In this paper we consider the following class of quasilinear elliptic problems: −div(a(x,∇u))=λh(x)exp(α0|u|nn−1)+f(x,u)inΩ, with the Dirichlet boundary condition where Ω⊂Rn(n≥2) is a smooth bounded domain and λ>0 is a positive parameter. We assume that there exists A:Ω×Rn→R such that a=∇A satisfies some mild conditions, h(x) and f(x,s) are mensurable functions and f(x,s) can enjoy exponential critical growth. The approach relies on a fixed point theorem and the Trudinger-Moser inequality.