CONCAVE-CONVEX QUASILINEAR ELLIPTIC PROBLEM SUBJECT TO A NONLINEAR BOUNDARY CONDITION

This paper deals with the existence of a positive solution to the problem { -Delta(p)u + u(p-1) = u(r-1) , x epsilon Omega, vertical bar del u broken vertical bar(p-2) partial derivative u/partial derivative v = lambda u(q-1,) x epsilon partial derivative Omega, where Omega subset of R-N is a bounded domain, v designates the unit outward normal to partial derivative Omega, Delta(p) is the p-Laplacian operator, 1 < q < p < r <= p* , p* = Np / (N - p) if p < N , p* = infinity otherwise, while lambda > 0. Our main result states the existence of Lambda > 0 so that positive solutions are only possible when 0 < lambda <= Lambda while the existence of a minimal positive solution is ensured in that range.