Multiple positive solutions for a superlinear elliptic problem in R2 with a sublinear Neumann boundary condition

Let Omega subset of R-2 be a bounded domain with C-2 boundary. Let u is an element of H-1(Omega) be a weak solution of the following problem: {-Delta u+u = p(u)e(u alpha)} u > 0 partial derivative u/partial derivative nu = lambda psi u(q) on partial derivative Omega in Omega, (P-lambda) where alpha is an element of(0, 2], lambda > 0, q is an element of [0, 1) and psi >= 0, a Holder continuous function on (Omega) over bar. Here p(u) is a polynomial perturbation of e(u alpha) as u --> infinity. Using variational methods we show that there exists 0 < Lambda < infinity such that (P-lambda) admits at least two solutions if lambda is an element of (0, Lambda), no solution if lambda > Lambda and at least one solution when lambda = Lambda. (c) 2006 Elsevier Ltd. All rights reserved.