The Existence Of Solutions For Two Classes Of Nonlinear Problem Via Critical Point Theory

Based on critical point theory, we considered two type of nonlinear problem in thisthesis: one is the existence of periodic solutions for Hamiltonian system, the other is theexistence of weak solutions for elliptic equation. More explicitly, we get the followingresults:1. Study the superquadratic second order Hamiltonian systems. A well-known openproblem raised by famous mathematician Lions Ekeland and Coti-Zelati in1991is:Whether the natural superquadratic condition is suffcient to ensure the existence of T-periodic solution for every T>0? This problem has not been solved for a long time. Thereason for this is that the functional correspond to system will not satisfy the （P.S.） con-dition at this time. Some extra condition is needed to recover the compactness whicheverapproximation methods or ordinary critical methods is used. By transfer the originalproblem into a problem defned on a fxed interval, the period T appears as a natural pa-rameter, which enable the usage of Struwe Technique. The structure information of thisproblem helps in recovering compactness, the bounded （P.S.） sequence can be obtaineddirectly for almost all parameter. This answer, although not entirely, the Lions-Ekeland-Zelati problem: the natural superquadratic condition is suffcient to ensure the existenceof-periodic solution for almost every T>0. Based on this theorem, two more theo-rem which ensure the existence of T-periodic solution for every T>0were obtained byadding extra conditions.Study the existence problem of periodic solutions for second order Hamiltonian sys-tems on prescribed energy surface. In eighties of last century, Gluck and Ziller, Benci andHayashi used totally different methods to prove the existence result with the conditionthat potential function is second order. This condition is critical for each method: themethod will not usable without this condition. With the help of Struwe technique, we areable to weaken this condition to frst order differential on almost every energy surface.The idea of transforming and using Struwe Technique in order to get compactnessfrom the structure of problem can also be used to study the subquadratic second orderHamiltonian systems, and the existence problem of weak solutions for elliptic equation.We obtained that the existence of T-periodic solution for almost every T>0with only natural subquadratic condition. On some specifc domain-spherical, we obtainedthe existence results for elliptic equation in almost every spherical under the natural su-perquadratic condition. These two results cover all the results before in autonomous case.2. Study asymptotically linear Hamiltonian systems which is resonance. Since theconditions about the behavior of potential function at infnity and at origin is weak, andthe functional correspond to the system is strong indefnite, we use Garlerkin approxi-mation methods. By careful analyzing the behavior of the approximated functionals atinfnity and origin, The critical groups of approximated functionals are computed and theexistence of critical point can be obtained by combining Morse theory and Maslov indextheory. Then periodic solutions ca be obtained by showing that the approximated solutionsequence formed by the critical points of approximated functional has convergent subse-quence. With this method the existence rusults were established under various situationof asymptotic matrix.3. Study discrete second order Hamiltonian systems. First we considered the ex-istence of infnitely many periodic solutions which has not been treated in the literature.Under an oscillating potential condition, two sequence of infnitely many periodic so-lutions was obtained: one sequence of solutions are local minimizers of the functionalcorresponding to the system, the other sequence are minimax type critical points of thefunctional. Then we consedered discrete second order Hamiltonian systems which isstrong resonance. The existence of at least two nontrivial periodic solution was obtained:one is the global minimum of the corresponding functional and the other is got by con-struct a contradiction.4. Study a nonsmooth functional which correspond to a p（x）-Laplacian equationwith a locally Lipschitz potential function, and prove its local minimum in1topologyis still a local minimum in W₀^1,p（x）（Ω） topology.