On Multiplicity of Positive Solutions for Quasilinear Equation with Co-normal Boundary Condition

Let Omega subset of IRN, N >= 3, be a bounded domain with C-2 boundary, p* = Np/N-p, the critical exponent for the Sobolev imbedding. In this work, we are interested in the following problem: (P-lambda) {-Delta(p)u + u(p-1) = u(p*-1) } in Omega u > 0 vertical bar del u vertical bar(p-2) partial derivative u/partial derivative v = lambda u(q) on partial derivative Omega, where lambda > 0, 0 <= q < p - 1. We show that there exists 0 < Lambda < infinity such that for suitable ranges of p and q, (P-lambda) admits at least two solutions in W-1,W-p (Omega) if lambda is an element of (0, Lambda) and no solution if lambda > Lambda. The proof of these assertions is done by first finding the local minimum for the variational functional associated to (P-lambda) and then applying mountain pass arguments to obtain a saddle point type solution. In the critical case we are considering, there are technical reasons which make the mountain pass argument work for only certain ranges of p and q.