Multiplicity of positive solutions to a p-Laplacian equation involving critical nonlinearity ✩

In this paper we deal with multiplicity of positive solutions to the p-Laplacian equation of the type −div |∇u|p−2∇u = μuq−1 + up∗−1, u∈W 1,p 0 (Ω), where Ω ⊂ RN is a bounded domain, N p2, 2 p q < p∗, p∗ = Np/(N − p), and μ is a positive parameter. We prove that there exists μ∗ > 0 such that, for each μ ∈ (0,μ∗), the equation has at least catΩ(Ω) positive solutions.  2003 Elsevier Science (USA). All rights reserved