Normalized Solutions For Two Classes Of Schrodinger-Poisson Equations

In this thesis,the existence and multiplicity of normalized solutions for two classes of Schrodinger-Poisson equations are investigated by using variational methods.Firstly,the existence and nonexistence of minimizers with prescribed L2-norm for a class of Schrodinger-Poisson equations are studied.Secondly,the multiplicity of normalized solutions for a class of Schrodinger-Poisson equations is considered.The main theoretical bases are the methods of the minimizing sequences,the compact embedding lemma,Ekeland's variational principle,the vanishing lemma,Pohozaev's identity and some analytical skills.Firstly,we consider the existence and nonexistence of minimizers for the following Schrodinger-Poisson equation#12where the paremeter ? is lagrange multiplier,??R+,N? 2,2<p<2*,2*=? if N=2 and 2*=2N/N-2 if N? 3,V and K satisfy the following conditions:#12(K)K ? L?(RN)\{0} is nonnegative function and(?)K(x)=K0>0.Suppose that ?(up to translations)is the unique positive radial solution for the following equation-?u+u-u1+4/N=0,u?H₁(RN)where N?2.Set ?*:=???24/N.The main result of the chapter is as follows.Theorem 2.1.1 Suppose that N? 2,V satisfies(V)and K satisfies(K),the following results hold:(1)If 2<p<2+4/N.then problem(P₁)has at least one minimizer for each ?>0.(2)If p=4/N+2,then problem(P₁)has minimizer if and only if 0<? ??*.(3)If p>4/N+2,then problem(P₁)has no minimizer for each ?>0.Secondly,we prove the multiplicity of normalized solutions for the following Schrodinger-Poisson equation(?)(P₂)where ? ? R,3 ?N ?5.The nonlinearity g:R?R is superlinear,subcritical,and possibly nonhomogeneous and satisfies the following conditions:(H₁)g:R?R is continuous and odd,(H₂)there exist ?,??R,2+4/N<???<4N-3/2(N-2)such that0<?G(s)?g(s)s ??G(s).K satisfies condition:(H₃)K ? L?(RN)\{0} is nonnegative function.The main result of the chapter is as follows.Theorem 3.1.1 Suppose that(H₁)-(H₃)hold.If c?1,then problem(P₂)has an unbounded sequence of pairs of radial solutions(±u_n,?_n)with ?v_n?22=c and ?_n<0 for each n ? N⁺.