Existence of positive solutions for a Schrödinger-Poisson system with critical growth

ABSTRACT

In this paper, we are concerned with the existence, multiplicity and concentration of positive solutions for the semilinear Schrödinger-Poisson system $\begin{array}{rl}& -{\epsilon }^2\mathrm{\Delta }u+a\left(x\right)u+\lambda \phi \left(x\right)u=b\left(x\right)f\left(u\right)+c\left(x\right)|u{|}^{4}u,x\in {\mathbb{R}}^3,\\ & -{\epsilon }^2\mathrm{\Delta }\phi =u^2,u\in H^1\left({\mathbb{R}}^3\right),x\in {\mathbb{R}}^3,\end{array}$ where $\epsilon >0$(small) and λ are real parameters, a,b and c are continuous functions, and f is a continuous superlinear and subcritical nonlinearity. First, we prove that there are two families of semiclassical positive solutions for $\epsilon >0$ small. Then we show that these solutions are concentrating on some sets which is generated by the minimal points of a and maximal points of b and c. In addition, we investigate the relation between the number of positive solutions and the topology of the set of the global minima(or maxima) of the potentials by minimax theorems and the Ljusternik-Schnirelmann theory. Finally, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.

Keywords:

• Positive solutions
• Schrödinger-Poisson system
• variational methods

2010 Mathematics Subject Classifications:

• 35J61
• 35J20
• 35Q55
• 49J40

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by National Natural Science Foundation of China [Grants 11571140, 11671077, 71704066], Fellowship of Outstanding Young Scholars of Jiangsu Province [BK20160063], the Six big talent peaks project in Jiangsu Province [XYDXX-015], and Natural Science Foundation of Jiangsu Province [BK20150478, BK20170542].