NONMINIMIZING POSITIVE SOLUTIONS FOR EQUATIONS WITH CRITICAL EXPONENTS IN THE HALF-SPACE

This paper is concerned with the existence of positive solutions of th e nonlinear elliptic problem -Delta u + a(x)u = u((N+2)/(N-2)), a(x) g reater than or equal to 0, with Neumann boundary conditions in a half- space Pi subset of R-N, N greater than or equal to 3. The main feature of the problem is a ''double'' lack of compactness due to the unbound edness of the domain and the presence of the critical Sobolev exponent . The solutions are searched using variational methods. although the f unctional related to the problem does not satisfy the Palais-Smale com pactness condition. We observe that the problem considered has no solu tions if a(x) is a positive constant; conditions on a(x) are given suf ficient to guarantee existence and multiplicity of positive solutions.