On the existence of solutions for nonhomogeneous Schrödinger-Poisson system

Boundary value problems

Shiwang Ma¹  Xiaoming Wang²  Lixia Wang³ 

[1] School of Mathematical Sciences and LPMC, Nankai University, Tianjin, China;School of Mathematics and Computer Science, Shangrao Normal University, Shangrao, China;School of Sciences, Tianjin Chengjian University, Tianjin, China

关键词: Schrödinger-Poisson systems;    sublinear nonlinearities;    concave and convex nonlinearities;    variational methods;    35B33;    35J65;    35Q55;   

DOI  :  10.1186/s13661-016-0584-9

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

In this paper, we study the existence of solutions for the following nonhomogeneous Schrödinger-Poisson systems:(∗){−Δu+V(x)u+K(x)ϕ(x)u=f(x,u)+g(x),x∈R3,−Δϕ=K(x)u2,lim|x|→+∞ϕ(x)=0,x∈R3,$$ (*) \quad \textstyle\begin{cases} -\Delta u +V(x)u+K(x)\phi(x)u =f(x,u)+g(x), &x\in\mathbb{R}^{3}, \\ -\Delta\phi=K(x)u^{2}, \qquad \lim_{|x|\rightarrow+\infty}\phi(x)=0, & x\in\mathbb{R}^{3}, \end{cases} $$wheref(x,u)$f(x,u)$is either sublinear in u as|u|→∞$|u|\rightarrow\infty$or a combination of concave and convex terms. Under some suitable assumptions, the existence of solutions is proved by using critical point theory.

【 授权许可】

CC BY