Existence Of Solutions To Nonhomogeneous Schrodinger-Maxwell System With Potential

Nonlinear phenomenon is very general in natural world, and is playing an important role in the natural science and engineering areas. Therefore, in recent years, people begin to pay more attention to the research of nonlinear systems, and various nonlinear systems have been discussed in many papers.For Schrodinger-Maxwell systems without the potential function and nonhomogeneous term, many researchers have conducted in-depth research, and get a lot of valuable results. However, for the systems with the potential function or nonhomogeneous term, particularly with both of them, the results of the current study are not comprehensive enough, because of the relatively complex nature of the systems’structure, thus we discuss such systems in this paper.This paper mainly consists of three chapters:Chapter 1 is introduction. Chapter 2 studies the existence of solutions to Schrodinger-Maxwell system with a bounded potential function V by using Sobolev imbedding theorem and Ekeland’s variational principle. In Chapter 3, we discuss the existence of solutions to Schrodinger-Maxwell system with an unbounded potential function and a positive coefficient.Consider the following Schrodinger-Maxwell system with a bounded potential function: where p ∈ (2,6),g∈ L2(M3), V∈C1(K3), and satisfy:(V) V ∈ C1(R3) and 1≤ V(x)≤V1<∞, x ∈ R3;(g) g ∈ C1(R3) ∩L2(R3) is nonnegative and radial function, satisfying 0<|g|2<Cp, where Cp is defined in Chapter 2.We use Holder’s inequality and Sobolev imbedding theorem to consider the correspond-ing functional, and discuss the convergence of the (PS) sequence, then look for the nontrivial solutions to the system.The following statement is the main result of this chapter:Theorem 2.3.1 Assume p ∈ (2,6), V,g satisfy (V), (g). Then there is a u0 ∈ Hr1(R3) such that I(u0)=inf{I(u):u ∈ Hr1(R3),||u||H1≤α}<0, and (u0,φu0) is a solution with negative energy to system (3), where φu0∈D1,2(R3) is the unique weak solution to-Δφ=u02, and I, α are defined in Chapter 2.In Chapter 3, we consider the following Schrodinger-Maxwell system with an unbounded potential function: where λ> 0,p ∈(2,6),g∈L2(R3), V ∈C1(R3).In order to obtain our statement, we need the following assumptions:(Vi) V ∈ C1 R3) satisfies inf x∈eR3 V(x)≥1, and for each M>0,m({x∈R3:V(x)≤ M})<∞, where m denotes the Lebesgue measure in R3;(g1) g ∈ C1(R3) ∩ L2(R3) is nonnegative, and g∈E satisfies 0<|g|2<CP/CE, where E, CE are defined in Chapter 3.We have the following conclusion:Theorem 3.2.1 Assume λ> 0,p ∈ (2,6), V,g satisfy (Vi), (g1). Then there is a u0 ∈ E such that Jλ(u0)=inf{Jλ(u):u ∈ E,||u||E ≤α}<0, and (u0,φu0) is a solution with negative energy to system (4), where φu0 ∈ D1,2(R3) is the unique weak solution to-Δφ=u02, and Jλ is defined in Chapter 3.