INFINITELY MANY WEAK SOLUTIONS FOR A p-LAPLACIAN EQUATION WITH NONLINEAR BOUNDARY CONDITIONS

We study the following quasilinear problem with nonlinear boundary conditions -Delta(p)u + a(x)vertical bar u vertical bar(p-2)u = f(x, u) in Omega, vertical bar del u vertical bar(p-2) partial derivative u/partial derivative nu = g(x, u) on partial derivative Omega, where Omega is a bounded domain in R-N with smooth boundary and partial derivative/partial derivative nu is the outer normal derivative, Delta(p)u = div(vertical bar del u|(p-2) del u) is the p-Laplacian with 1 < p < N. We consider the above problem under several conditions on f and g, where f and g are both Caratheodory functions. If f and g are both superlinear and subcritical with respect to u, then we prove the existence of infinitely many solutions of this problem by using "fountain theorem" and "dual fountain theorem" respectively. In the case, where g is superlinear but subcritical and f is critical with a subcritical perturbation, namely f(x, u) = vertical bar u vertical bar(p*-2)u + lambda 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 1 < r < p, by using "mountain pass theorem" and "concentration-compactness principle" respectively.