Infinitely Many Small Energy Solutions Of Two Classes Of Strongly Indefinite Problems

In this thesis, based on variational methods and critical point theory, we consider the existence of infinitely many small energy solutions of two class of strongly indefinite problems. By choosing suitable spaces and functionals, we get the existence of critical points for the functional with the generalized fountain theorem which is aimed at the strongly indefinite problems, so the existence criteria of solution of the corresponding strongly indefinite problems are established.First, we introduce the background of our problems and its recent devel-opment. The main results and method of this dissertation are given, and some preliminary knowledge are also listed.Then, we consider the existence of infinitely many small energy solutions for the following Schrodinger equation with concave and convex nonlinearities when the weight vanishes quickly enough, we can get the (PS)-condition with compact embeddings. Otherwise, we get the nontriviality of solutions with the concentration-compactness principle. Then sufficient condition of the existence infinitely many small energy solutions is given by using the dual form of the generalized fountain theorem.Finally, we study the existence of infinitely many small energy homoclinic solutions for the first order Hamiltonian system when and H is of the following form:Where the nonlinearity is also with a weight, which is different from that in the literature. Using the dual form of the generalized fountain theorem, we established the sufficient condition of the existence of the existence of infinitely many small energy homoclinic solutions.