Solutions for critical elliptic systems on compact manifolds

Abstract

The work presented in this thesis addresses results concerning the existence of solutions for three classes of strongly coupled elliptic systems on compact Riemannian manifolds without boundaries. In these classes, coupled nonlinearities with critical exponents in the sense of Sobolev and Hardy-Sobolev embeddings are involved. The first and second classes of problems involve the Laplace-Beltrami operator on a manifold and nonlinearities with a critical Sobolev exponent in the first case and Hardy-Sobolev exponent in the second case. In the second class, we also consider Hardy-type potentials. The third problem involves the p-Laplacian operator and a nonlinearity with a critical Hardy-Sobolev exponent. Thus, in both problems, we investigate the lack of compactness and how to recover it at some energy level. In this work, the approach is conducted through variational methods.