Existence Of Solutions For Semilinear Elliptic Equations And Hamiltonian Systems

This paper first briefly introduces the contents , status and meaning of the study of nonlinear partial differential equations, then we consider the existence of multiple solutions for a class of second order semilinear elliptic equations; existence of nontrivial solutions for nonlinear biharmonic equations; existence of periodic solutions for nonlinear Hamiltonian systems. It can be divided into the following chapters:(1) The first chapter of this thesis is the introduction. First, we briefly introduce the contents , status and meaning of the study of nonlinear partial differential equations, which contain the current research at home and abroad as well as the acquisition of relevant achievements, and the definition of nonlinear partial differential equations. Then the second section briefly introduces the important role of the minimax theory in the proof of the existence of (multiple) solutions of nonlinear partial differential equations, and given two minimax theorems: mountain pass theorem and linking theorem. Last the third section given two minimax theorems which are used in this paper:"a variation of linking"theorem and"linking scale"theorem.(2) In the second chapter, we consider the existence of multiple solutions for a class of second order semilinear elliptic equations by"a variation of linking"theorem and"linking scale"theorem. This chapter is divided into three sections. The first one is prior knowledge, we introduce some propertys, definitions and theorems. In the second section, we consider the relationship between multiplicity solutions and the nonlinear term of second order semilinear elliptic equations (2-1) in [25]. We prove that the equation has two nontrivial solutions by a variation of linking theorem withλk <λ<λk+1. Moreover, it prove that the equation has three nontrivial solutions by linking scale theorem.The last one is summary.(3) In the third chapter, we consider the existence of nontrivial solutions for nonlinear biharmonic equations. This chapter is divided into three sections. The first one is prior knowledge, we introduce some propertys, definitions and theorems. The second one prove that the equation has at least one nontrivial solution by mountain pass theorem, i.e. it has at least two solutions. Moreover, it prove that the equation has at least two nontrivial solution by a variation of linking theorem, i.e. it has at least three solutions. The last one is summary.(4) In the fourth chapter, we consider the existence and multiplicity results for some fourth order semilinear elliptic problems. This chapter is divided into three sections. The first one is prior knowledge, we introduce some propertys, definitions and theorems. The second one prove that the equation has at least one nontrivial solution by mountain pass theorem, i.e. it has at least two solutions. Moreover, it prove that the equation has at least two nontrivial solution by a variation of linking theorem, i.e. it has at least three solutions. The last one is summary.(5) In the fifth chapter, we consider the existence of periodic solutions for nonlinear Hamiltonian systems. Now, we made some improvements for theorem 6.10 in [25], and get that Hamiltonian systems has two nonconstant periodic solution by"a variation of linking"theorem.