A class of quasilinear elliptic equations with singular and exponential terms

Abstract

We use a scheme of approximation together with Brouwer's theorem to establish the existence of positive solutions for a class of quasilinear elliptic problems. More precisely, we study the class of equations: (1) { − Div ( a ( | ∇u | p ) | ∇u | p − 2 ∇u ) = η u − γ + λ u s + f ( u ) in Ω , u = 0 on ∂Ω , where Ω ⊂ R N ( N ≥ 3 ) is a bounded domain with smooth boundary, 2 ≤ p < N , 0 < γ < 1 , 0<s<N−1, η and λ are positive parameters, a : R + → R + is a function of class C 1 ( R + ) . The nonlinear term f involves three different types of Trudinger–Moser growth: subcritical, critical or supercritical. We also study the asymptotic behavior of the solutions with respect to the parameters.