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 ) ) = λ h ( x ) exp ( α 0 | u | n n − 1 ) + f ( x , u ) in Ω , with the Dirichlet boundary condition where Ω ⊂ R n ( n ≥ 2 ) is a smooth bounded domain and λ > 0 is a positive parameter. We assume that there exists A : Ω × R n → 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.