Positive Solutions Of Asymptotically Linear Schrodinger-Poisson Systems

In this paper,we study the following Schrodinger-Poisson system: where the parameterλ∈(0,+∞)and assume that the following conditions:(2) There exists l∈(0,+∞) such that (?)f(s)/s=l;(3)V∈C(R3,R)∩L∞(R3,R)and V(x)≥V0>0for all x∈R3；(4)q∈L2(R3)\{O),q(x)≥0for all x∈R3；(5) By the Mountain Pass Theorem,we can obtain the following result:Theorem2.1.1.Suppose that(1)一(5)hold.Then there exists λ0>0,such that system(SP)has a positive ground state solution for any λ∈(0,λ0).We study the following nonhomogeneous Schrodinger-Poisson system: where the parameter λ∈(0,+∞),(1)-(5)hold and assume that the following condition:(h)h∈L2(R3)\{0} and h(x)≥0,x∈R3.By the Mountain Pass Theorem,we can obtain the following result:Theorem2.1.2. Suppose that(1)一(5)and(h)hold. Then there exist λ1>0and m>0,if||h||L2(R3)<m,then system(H)has two positive solu-tions(u1,φ),(u2,φ)satisfying Jλ(u1)<0and Jλ(u2)>0for any λ∈(0,λ1).