Modified quasilinear equations with strongly singular and critical exponential nonlinearities

Abstract

In this paper, we study global multiplicity result for a class of modified quasilinear singular equations involving the critical exponential growth:where Ω is a smooth bounded domain in R 2 , 0 < q < 3 and α : Ω → (0, +∞) such that α ∈ L ∞ (Ω).The function f : Ω × R → R is continuous and enjoys critical exponential growth of Trudinger-Moser type. Using a sub-super solution method, we show that there exists some Λ * > 0 such that for all λ ∈ (0, Λ * ) the problem has at least two positive solutions, for λ = Λ * , the problem achieves at least one positive solution and for λ > Λ * , the problem has no solution.