On a Neumann problem with supercritical exponent and supercritical nonlinearity on the boundary

Let Omega be a smooth bounded domain of R(n), with n greater than or equal to 3, a (x), f (x) two functions of C-infinity (<(Omega)over bar>) and b (x), h (x) two functions of C-infinity (partial derivative Omega). We study the existence and the multiplicity of positive solutions and of nodal solutions for the equation Delta u + a (x) u = f (x) \u\(p-2) u on Omega with partial derivative u/partial derivative v + b (x) u = h(x) \u\(q-2) u on partial derivative Omega. Symmetry assumptions allow us to take supercritical values p greater than or equal to 2 n/(n - 2) and q greater than or equal to (2n - 2)/(n - 2) of the exponents.