Multiple positive solutions for a quasilinear elliptic equation with critical exponential nonlinearity

Let ΩΩ be a bounded domain in RN,N≥2RN,N≥2, with C2C2 boundary. In this work, we study the existence of multiple positive solutions of the following problem: (Pλ){−div(|∇u|2N−21+|∇u|2N∇u)+|u|2N−21+|u|2Nu=upeuNN−1,u>0inΩ,|∇u|2N−21+|∇u|2N∂u∂ν=λψuqon∂Ω, where λ>0,p>2N−1,q∈(0,N−1)λ>0,p>2N−1,q∈(0,N−1) and ψψ is a non-negative Hölder continuous function. We show that there exists Λ∈(0,∞)Λ∈(0,∞) such that (Pλ)(Pλ) admits at least two solutions in W1,N(Ω)W1,N(Ω) for λ∈(0,Λ)λ∈(0,Λ) and no solution for λ>Λλ>Λ.