Existence And Relative Dynamical Property Of Solutions For Some Nonlinear Elliptic Dynamic Systems In Whole Space

The concept of nonlinear dynamical systems originated from the research about thephenomenon of oscil1tion, turbulence and nudism movement in the last nineteen century.Experiencing one hundred years development, nonlinear dynamical systems have madegreat achievements. The theory of nonlinear dynamical systems has various applicationsnot only in science and engineering (especially in equipment manufacturing, railway ve-hicle system design, electro-mechanical coupled system, power system, spacecraft controlsystem, microcosmic system and the seepage in fissured rocks), but also in economical andsocial area.This dissertation will be concerned about some classes of nonlinear elliptic dynami-cal systems. They are the most basic nonlinear dynamical systems in microcosmic system(quantum mechanics) which are as important as Newton equation in classical mechanics.Under proper assumptions, we study some elliptic dynamical systems and obtain the exis-tence and multiplicity of solutions for the corresponding systems and further study someproperties such as concentration etc by using variational method. To some extent, we im-prove some known results in this dissertation.Chapter1of the dissertation is devoted to the introduction, which briefly gives thebackground of nonlinear elliptic dynamical systems, introduces some preliminaries of vari-ational method and shows the main results of this thesis. From Chapter2of this disser-tation, the four classes of elliptic dynamical systems in dynamics are deeply investigatedand some improvements are obtained.The first class of system that we investigate is the singularly coupled elliptic systemunder the effect of general nonlinearities. This content is contained in Chapter2. In thischapter, we not only obtain the existence of solutions but also study the concentrationphenomena to the solutions. The results of this chapter explore what are the essentialfeatures which guarantee the existence of localized ground state solutions.The second class of system that we are concerned with is a critical singular ellipticsystem in whole space with concave and convex nonlinearity and sign-changing weightfunction. In recent years, much attention has been paid to the investigation of this ellip-tic system in bounded domains. Up to now, there are few papers on this problem inRN.The purpose to Chapter3is to investigate the existence and a multiplicity of positive solu-tions of the elliptic system by using the decomposition of the Nehari manifold. This workimproves the known results. The third class of system considered by this thesis is the perturbed nonlinear ellipticsystem with electromagnetic fields. This part contains Chapter4and Chapter5. Motivatedby these known results, Chapter4and Chapter5consider the existence and multiplicity ofsemiclassical solutions to the elliptic equation and elliptic system in whole space involvingcritical nonlinearities and magnetic field. The main difficulty is to overcome the loss of thecompactness. The corresponding results of Chapter4and5are an obvious extension orimprovement to the known results.The fourth class of system that we study is a p-Laplacian system inRNinvolving crit-ical nonlinearity. Chapter6studies the existence of nontrivial solutions for the p-Laplaciansystem inRNwith critical nonlinearity. The main difficulty in the case is the lack of com-pactness of the energy functional associated to the system because of unbounded domainRNand critical nonlinearity. To overcome this difficulty, we prove the energy functionalpossesses (P S)ccondition at some energy level c. Furthermore, we get the existence re-sult by using the mountain pass theorem due to Rabinowitz. The main results of Chapter6can be seen as a complement or improvement of the results developed in Garcia andAzorero[1987], Ding and Lin[2007] and Alves and Ding[2003].