Existence And Multiplicity Of Solutions For Schrodinger-Poissoil System

This paper is mainly composed of two chapters. Chapter1is the introduction. Chapter2studies the existence and multiplicity to solutions for Schrodinger-Poisson system in using the mountain pass theorem and the fountain theorem respectively.In chapter2, the following Schrodinger-Poisson system is considered, where f∈C(R3×R,M) and V∈C(R3,R) satisfy(V1) V satisfies inf x∈R3V(x)≥V0>0and for each M>0, m({x∈R3: V(x)≤M})<∞, where m({x∈R3: V{x)≤M} denotes the Lebesgue measure in R3;(f1) There exist c>0,p∈(4,6) such that|f(x,t)|c(|t|p-1+1), x∈R3,t∈R, and f(x,t)t≥0for t≥0;(f2) There exists a∈[0, V0) such that limsup f(x,t)/t <a/2, uniformly for a.e. x∈R3;(f3) lim|t|â†'∞F(x, t)/t4=∞uniformly for a.e. x∈R3, with F(x, t)=f0t f(x, s)ds;(f4) There exists σ∈[0, V0),p∈(4,6) such that f(x,t)t-pF(x,t)≥-σt2, x∈R3,t∈R;(f5)/(x,-t)=-f(x, t) for all (x, t)∈R3×R;(f6) There exists p∈(4,6) such that h(s)=F(x,s-1t)sp is not increasing;(f7) infx∈R3,|t|=1F(x,t)>0.The following theorems are the main results of this paper.Theorem2.1.1Suppose V,f, λ>0satisfy these conditions Vi,(f1)-(f5), λ=1, then problem (1) has infinitely many solutions{(uk,φk)} such that when kâ†'∞, we haveTheorem2.1.2Under the assumptions V1,(f1),(f2),(f4),(f6),(f7), the problem (1) has at least one nontrivial solution provided that A>0is sufficiently small.