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)=ηu−γ+λus+f(u)inΩ,u=0on∂Ω,(1) where Ω⊂RN (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 C1(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.