Nontrivial Solutions For Two Classes Of Nonlinear Schr(?)dinger Equations

In this thesis,the existence of nontrivial solutions for two kinds of nonlinear Schr(?)dinger equations is investigated by using variational methods.Firstly,the existence and the nonex-istence of constrained minimizers for a class of nonlinear Schr(?)dinger-Poisson equations are studied.Secondly,the existence of weak solution for a class of nonlinear eigenvalue problem-s is considered.The main theoretical bases are the methods of the minimizing sequences,Ekeland's variational principle,the vanishing lemma,the mountain pass lemma,the minimax method and some analytical skills.In chapter 2,the following Schr(?)dinger-Poisson equation(?) is investigated,where the space dimension N? 2,the paremeter ? is lagrange multiplier,p>0,if N=2 and 0<p<4/N-2,if N>3.V and a satisfy the following assumptions:(?) Assume that a*=??(x))?24/N,where ?(x)is the radial solution for the following equation-?u+u=u1+4/N,u?H1(RN).The main result of the chapter is as follows.Theorem 2.1.1 Assume that V satisfies(V) and a satisfies(A),we have(1)if 0<p<4/N,then the problem(P1) has at least one minimizer.(2)if p=4/N and a0>a*,then the problem(P1) has no minimizer,and if a??a*,then problem(P1)has minimizer.(3)if p>4/N,then the problem(P1) has no minimizer.In chapter 3,the following nonlinear eigenvalue equation(?) is considered,where ??R,ai>0,3/2<pi<7/4,i=1,2,…,m.The main result of the chapter is as follows.Theorem 3.1.1 The equation(P2) has weak solutions(u,?)? H1(R4)× R,with(?)R4 |u|2dx=1 and ?<0.