Positive solutions for some quasilinear equations with critical and supercritical growth

We establish results concerning the existence and multiplicity of positive solutions for the problem -div(a(epsilon x) vertical bar del u vertical bar(p-2)del u) + u(p-1) = f(u) + u(p*-1) in R-N, u is an element of W-1,W- p(R-N), where epsilon > 0 is a small parameter, 2 <= p < N, p* = Np/(N - p), a is a positive potential and f is a superlinear function. We obtain the existence of a ground state solution and relate the number of positive solutions with the topology of the set where a attains its minimum. We also prove a multiplicity result for a supercritical version of the above problem. In the proofs we use minimax theorems and Ljusternik-Schnirelmann theory. (c) 2006 Elsevier Ltd. All rights reserved.