Existence And Properties Of Solutions For Elliptic Equation (System) Involving Critical Growth In Whole Space

Variational method is one of the important fundamental method in the nonlinear functional analysis. The basic idea is that the problem of differential equations convert to the critical point of corresponding functional problem. This dissertation will be concerned about some classes of nonlinear elliptic equations(system) by using variational method. These type equations(system) in physics and mechanics have a strong practical application backgrounds. Under proper assumptions, we will study some classes of nonlinear elliptic equations(system) and obtain the existence or multiplicity of solutions for the corresponding equations(system) and further study some properties such as attenuation, asymptotic property. To some extent, we improve some known results in this dissertation.There are five chapters in the dissertation. Chapter 1 of the dissertation is devoted to the introduction, which briefly gives the background of nonlinear elliptic equations(system), introduces some preliminaries of variational method, research status and shows the main results of this thesis. From Chapter 2 of this dissertation, the four classes of elliptic equations(system) are investigated. In the past researches, single perturbed Schr ¨odinger or nonlinear subcritical perturbed Schr ¨odinger system are considered. In this dissertation, we will study critical equations(system) in whole space,so we obtain some improvements.The first class of system that we investigate is the nonlinear perturbed elliptic system. This content is contained in Chapter 2. In the last references, many researchers studied single perturbed Schr ¨odinger equation and obtained existence of the least energy solution. Some papers considered the existence of positive solution to singular perturbed elliptic system on bounded domain. In this chapter, we obtain the existence of solutions of two type perturbed elliptic system in whole space. The results of this chapter improves the known results.The second class of system that we are concerned with is a critical perturbed elliptic system in whole space with electromagnetic fields. This part is included in Chapter 3. In recent years, much attention has been paid to the investigation of single elliptic equation under subcritical with special nonlinear term. Some improvements are obtained in this chapter.The third class of system considered by this thesis is the nonlinear Schr ¨odingerPoisson system by using extension Clark’s theorem. This content is contained in Chapter 4. Motivated by these known results, Chapter 4 not only obtain the existence of solutions to Schr ¨odinger-Poisson system in whole space but also study the attenuation to the solutions. In former literature, the state of attenuation is few mentioned.The fourth class of system that we study is a Kirchhoff-type equation in RNinvolving critical nonlinearity. Chapter 5 studies the existence of nontrivial solutions for Kirchhoff-type equation in RNwith critical nonlinearity and discusses the asymptotic property of the solution. The main difficulty in the case is the lack of compactness of the energy functional associated to the system because of unbounded domain RN and critical nonlinearity. To overcome this difficulty, we prove the energy functional possesses(P S)ccondition.