Nonminimizing positive solutions for equations with critical exponents in the half-space

This paper is concerned with the existence of positive solutions of the nonlinear elliptic problem -Delta u + a(x)u = u((N+2)/(N-2)), a(x) greater 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 unboundedness of the domain and the presence of the critical Sobolev exponent. The solutions are searched using variational methods. although the functional related to the problem does not satisfy the Palais-Smale compactness condition. We observe that the problem considered has no solutions if a(x) is a positive constant; conditions on a(x) are given sufficient to guarantee existence and multiplicity of positive solutions.