Multiple solutions for the p&q-Laplacian problem with critical exponent

Abstract

In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p&q-Laplacian type involving the critical Sobolev exponent: {−Δpu − Δqu = |u|p*−2u+μ|u|r−2u in Ω,u|∂Ω = 0,where Ω ⊂ ℝN is a bounded domain, N > p, p* = NpN−p is the critical Sobolev exponent and μ > 0. We prove that 1 <r <q <p <N, then there is a μ0 > 0, such that for any μ ∈ (0,μ0), the above mentioned problem possesses infinitely many weak solutions. Our result generalizes a similar result in [8] for p-Laplacian type problem.