The Research Of Some Topics Of Ground State Solutions And Concentration For Elliptic Type Equations

In this dissertation, we study some topics of ground state solutions and concen-tration for elliptic type equations. We focus on Neumann boundary value problems,fractional Schrodinger equations, Schrodinger-Poisson systems and Kirchhoff prob-lems. The main results are as follows:In Chapter 1, we review some notations and conventions, and give some useful preliminary results which will be used in the later chapters.In Chapter 2, we are concerned with a sublinear Neumann problem. In contrast to previous work on the Dirichlet problem, some difficulties arise due to the fact that the associated energy functional is not bounded from below. Complementing recent work by Parini and Weth in [93] on least energy solutions, we prove that this Neumann problem has infinitely many solutions with small negative energy.In Chapter 3, we study the existence, nonexistence and mass concentration of L2-normalized solutions for nonlinear fractional Schrodinger equationsu Comparing to previous work on the Schrodinger equation, we encounter some new challenges due to the nonlocal nature of the fractional Laplacian. We first prove that the optimal embedding constant for the fractional Gagliardo-Nirenberg-Sobolev inequality can be expressed by exact form, which improves the results of [57, 58]. By doing this,we then establish the existence and nonexistence of L2-normalized solutions for this equation. Finally, under a certain type of trapping potentials, by using some delicate energy estimates we present a detailed analysis of the concentration behavior of L2-normalized solutions in the mass critical case.In Chapter 4, we are concerned with Schrodinger-Poisson systems. Due to its relevance in physics, the system has been extensively studied and is quite well under-stood in the three dimensional case. In contrast, much less information is available in the planar case which is the focus of the present chapter. It has been observed by Cingolani and Weth [36] that the variational structure of Schrodinger-Poisson systems differs substantially in the planar case and leads to a richer structure of the set of solu-tions. However, the variational approach of [36] is restricted to the case p > 4 which excludes some physically relevant exponents. In the present chapter, we remove this unpleasant restriction and explore the more complicated underlying functional geom-etry in the case 2 < p < 4 with a different variational approach.In Chapter 5, we are concerned with a class of Kirchhoff equations. Under appro-priate assumptions, we prove the existence of ground state solutions for this equations by using variational methods. Furthermore, we also investigate the phenomenon of concentration of ground state solutions.