SINGULAR ELLIPTIC EQUATIONS WITH NONLINEARITIES OF SUBCRITICAL AND CRITICAL GROWTH

In this paper, we consider the problem -Delta u = -u(-beta) chi({u) (>) (0}) + f(u) in Omega with u = 0 on partial derivative Omega, where 0 < beta < 1 and Omega is a smooth bounded domain in R-N, N >= 2. We are able to solve this problem provided f has subcritical growth and satisfy certain hypothesis. We also consider this problem with f(s) = lambda s + s(N+2/N-2) and N >= 3. In this case, we are able to obtain a solution for large values of lambda. We replace the singular function u(-beta) by a function g(is an element of)(u) which pointwisely converges to u(-beta) as is an element of -> 0. The corresponding energy functional to the perturbed equation -Delta u +g(is an element of)(u) = f(u) has a critical point u(is an element of) in H-0(1)(Omega), which converges to a non-trivial non-negative solution of the original problem as is an element of -> 0.