Research On The Existence Of Solutions For Schr(?)dinger-poisson Equations And Kirchhoff Type Equations

In this dissertation, we study the positive solutions of two types of nonlocal elliptic partial differential equations by using variational methods. The first kind is Schr(?)dinger-Poisson equations coming from quantum electrodynamics and semiconductor theory and so on, which, as a model, is used to describe any state of a particle in the 3-dimensional time-space. Indeed, in the presence of many particles one can try to simulates the effects of the mutual interactions by introducing a nonlinear term. One also introduces the Poisson potential to describe the interaction between a charge particle interacting with the electromagnetic field. When the nonlinear term has critical growth, we obtain some results on the existence of positive ground state solutions and semiclassical state solutions. Moreover, we also prove the concentration and exponential decay of semiclassical state solutions when the the Plank’s constant converges to zero. The second type is Kirchhoff type equations which takes into account the changes in length of the string produced by transverse vibrations. And it also be widely used in the fields of non-Newtonian mechanics, cosmic physics, blood plasma, elastic theory, population dynamics and so on. We study the existence and multiple of positive ground state solutions and semiclassical state solutions. The dissertation consists of five chapters. The main contents are as follows:In chapter 1, the historical background, status and the up-to-date progress for all the investigated problems are introduced, the main contents of the dissertation are outlined, and some preliminary tools used in the proof of main results are given.In chapter 2, by means of an abstract critical point theorem and concentrationcompactness principle, we study the existence of positive ground state solutions for a class of Schr(?)dinger-Poisson equations with critical growth. We have to analyze the limit version of such a class of equations without any compactness conditions.We firstly make use of concentration-compactness principle to prove that the limit equations has at least a positive ground state solution in a nature constraint manifold. Then we investigate the behavior of the Palais Smale sequences and establish a local splitting compactness theorem. At last, combining the abstract critical point theorem with the above facts, we get the existence of solutions of original equations.In chapter 3, without assuming the monotonicity of the function t â†'f(t)/t3, we investigate the existence of the semiclassical state solutions for a class of chr(?)dingerPoisson equations with a general critical growth. Since the linear potential term of such a class of equations may be coercive, the standard Sobolev space does not work in our consideration. We firstly use penalization approach to prove that the modified equation has a mountain pass solution and then carefully study the behavior of the limit version of the original equations. We also establish a new local splitting compactness which help us to obtain that the original equation has at leat a positive bounded state solution. We obtain, by the maximum principle that positive bounded state solution has a exponential decay.In chapter 4, without assuming the monotonicity of the function t â†'f(t)/t3, we study the existence of positive ground state solutions for Kirchhoff-type problem with a general nonlinearity in the critical growth. By restricting domain of the corresponding energy functional at a abstract Pohozave manifold, we can employ the behavior of compact embedding in the space of radial functions to get convergence of minimizing sequence. Using Pohozaev type identity, we also show that the mountain pass value gives the least energy level.In chapter 5, we focus on the multiplicity, concentration and exponential decay of semiclassical state solutions of the Kirchhoff type equations with critical exponent growth. Under some suitable conditions, we show that the number of positive solutions depends on the profile of the linear potential term. Moreover, we also prove the existence of positive ground state solutions. The outline of our arguments is that, for each of strict global minimum points of the linear potential function,we define a Nehari solution manifold and then use a generalized barycentre map to differentiate such some manifolds. Combining concentration and compactness principle with the Ekeland variational principle to prove the convergence of the minimizing sequences.