CRITICAL GROWTH ELLIPTIC PROBLEM IN R2 WITH SINGULAR DISCONTINUOUS NONLINEARITIES

Let Omega be a bounded domain in R-2 with smooth boundary, a > 0, lambda > 0 and 0 < delta < 3. We consider the following critical problem with singular and discontinuous nonlinearity: -Delta u = lambda(chi({u<a})u(-delta) + h(u)e(u2)) in Omega, u > 0 in Omega, u = 0 on partial derivative Omega, where chi is the characteristic function and h(u) is a smooth nonlinearity that is a "perturbation" of e(u2) as u -> infinity (for precise definitions, see hypotheses (H1) - (H5) in Section 1). With these assumptions we study the existence of multiple positive solutions to the above problem.