MULTIPLE POSITIVE SOLUTIONS FOR A SINGULAR ELLIPTIC EQUATION WITH NEUMANN BOUNDARY CONDITION IN TWO DIMENSIONS

Let Omega subset of R-2 be a bounded domain with C-2 boundary. In this paper, we are interested in the problem -Delta u + u = h(x, u)e(u2)/vertical bar x vertical bar(beta), u > 0 in Omega, partial derivative u/partial derivative v = lambda psi u(q) on partial derivative Omega, where 0 is an element of partial derivative Omega, beta is an element of [0, 2), lambda > 0, q is an element of [0, 1) and psi >= 0 is a Holder continuous function on (Omega) over bar. Here h(x, u) is a C-1((Omega) over bar x R) having superlinear growth at infinity. Using variational methods we show that there exists 0 < Lambda < infinity such that above problem admits at least two solutions in H-1(Omega) if lambda is an element of (0, Lambda), no solution if lambda > Lambda and at least one solution when lambda = Lambda.