A priori estimates or super-linear elliptic equation: the Neumann boundary value problem

In this paper we study the nonexistence of finite Morse index solutions of the following Neumann boundary value problems (Eq.H){-Delta u = (u(+))(p) in R-+(N), partial derivative u/partial derivative x(N) = 0 on partial derivative R-+(N), u is an element of C-2 (<(R-+(N))over bar>) and sign-changing, u(+) is bounded and i(u) < infinity (Eq.H'){-Delta u = vertical bar u vertical bar(p-1)u in R-+(N), partial derivative u/partial derivative x(N) = 0 on partial derivative R-+(N), u is an element of C-2 (<(R-+(N))over bar>), u is bounded and i(u) < infinity. As a consequence, we establish the relevant Bahri-Lions's L-infinity-estimate [3] via the boundedness of Morse index of solutions to {-Delta u = F(x,u) in Omega, partial derivative u/partial derivative v = 0 on partial derivative Omega, where f has an asymptotical behavior at infinity which is not necessarily the same at +/-infinity. Our results complete previous Liouville type theorems and L-infinity -bounds via Morse index obtained in [3, 6, 13, 16, 12, 21].