Several Applications Of Critical Point Theory And Drop - Invariant Set

In this paper, we mainly consider two problems. Firstly, we concern with the existence of multiple solutions, and sign changing solutions to the class of Kirchhoff type equations: where Ω is smooth bounded domain in R3, and a, b> 0. In our paper, we amuse that the nonlinear term f(x,u) is not satisfying P.S. condition. However, we have found a more interesting and weaker condition. Under the weaken condition, we can still obtain at least one nontrivial solution via mountain pass theorem. If the nonlinear term f(x,u) is odd, we can obtain an unbounded solution sequence by fountain theorem. Moreover, we can obtain at least one positive, one negative and one sign-changing solutions by invariant sets of descent flow theory. Although we have relaxed restrictions on the conditions, we can still enrich our results.Secondly, we consider the following unbounded Hamiltonian system: where H E C1(R×R2M× RN; R). Unlike previous work, in our case the essential spectrum family may contain 0. We give an extension of the nonlinear terms which does not satisfies the general P.S. condition. Based on the local linking theory, we can still obtain at least one nontrivial solution.