Multiple positive solutions for a kind of singular Schrödinger–Poisson system

ABSTRACT

In this paper, we discuss the following couple of Schrödinger–Poisson system with singularity: $\left\{\begin{array}{ll}-\mathrm{\Delta }u-\phi u=\lambda h\left(x\right)u+\frac{\mu }{|x{|}^{\beta }{u}^r}+u^3,& \mathrm{in}\mathrm{\Omega },\\ -\mathrm{\Delta }\phi =u^2,& \mathrm{in}\mathrm{\Omega },\\ u>0,& \mathrm{in}\mathrm{\Omega },\\ u=\mathrm{\Phi }=0,& \mathrm{on}\mathrm{\partial \Omega },\end{array}\right.$ where $\mathrm{\Omega }\subset {\mathbb{R}}^3$ is a bounded smooth domain with boundary $\mathrm{\partial \Omega }$, μ, $\lambda >0$ are parameters, $r\in (0,1)$ and $\beta \in (0,(5+r\left)/2\right)$ are constants. The existence and multiplicity of positive solutions for this system are considered under some suitable conditions by Nehari manifold.

Keywords:

• Schrödinger–Poisson system
• singularity
• multiplicity

MSC (2010):

• 35J55
• 35J60
• 35J47
• 35J20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Scientific Research Projects for the Education Department of Jilin Province (grant. JJKH20180859KJ).