Existence of infinitely many weak solutions for the p-Laplacian with nonlinear boundary conditions

In this paper we deal with the existence of weak solutions for quasilinear elliptic problem involving a p-Laplacian of the form [GRAPHICS] We consider the above problem under several conditions on f. For f "superlinear" and subcritical with respect to to, we prove the existence of infinitely many Solutions of the above problem by using the "fountain theorem" and the "dual fountain theorem" respectively. For the case where f is critical with a subcritical perturbation, namely f(x, u) = vertical bar u vertical bar(p)*(-2) + vertical bar u vertical bar(r-2)u, we show that there exists at least a nontrivial solution when p < r < p* and there exist infinitely many Solutions when l < r < p, by using the "mountain pass theorem" and the "concentration-compactness principle" respectively. (c) 2008 Published by Elsevier Ltd