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Variational problems involving non-local elliptic operators (Ph.D. Thesis)
ABSTRACT. This thesis deals with nonlinear elliptic problems involving a nonlocal integrodifferential operator of fractional type. Our main results concern the existence of weak solutions for these problems and they are obtained using variational and topological methods.
In the first part of this thesis we consider non-local fractional problems with nonlinarities satisfying an aysmptotically linear growth at infinity. In some cases we are able to treat both the non-resonant case and the resonant case.
The second part is devoted to the study of fractional Kirchhoff problems with a critical nonlinearity. The main difficulty of this kind of problems is the verification of compactness condition. However we give two possible proofs of this condition.
The existence theorems obtained here extend some results, well known for the classical Laplace operator, to the non-local fractional setting.