# The existence and concentration of positive ground state solutions for a class of fractional Schr\"{o}dinger-Poisson systems with steep potential wells

## TL;DR

This paper proves the existence and concentration of positive ground state solutions for a class of fractional Schrödinger-Poisson systems with steep potential wells, using new analytical methods and without requiring monotonicity of the nonlinearity.

## Contribution

It establishes the existence and concentration of solutions for fractional Schrödinger-Poisson systems with steep potential wells, removing the need for monotonicity assumptions on the nonlinearity.

## Key findings

- Existence of positive ground state solutions.
- Concentration behavior of solutions as parameter varies.
- Applicability to systems with non-monotone nonlinearities.

## Abstract

The present study is concerned with the following fractional Schr\"{o}dinger-Poisson system with steep potential well: $$ \left\{% \begin{array}{ll} (-\Delta)^s u+ \la V(x)u+K(x)\phi u= f(u), & x\in\R^3, (-\Delta)^t \phi=K(x)u^2, & x\in\R^3, \end{array}% \right. $$ where $s,t\in(0,1)$ with $4s+2t>3$, and $\la>0$ is a parameter. Under certain assumptions on $V(x)$, $K(x)$ and $f(u)$ behaving like $|u|^{q-2}u$ with $2<q<2_s^*=\frac{6}{3-2s}$, the existence of positive ground state solutions and concentration results are obtained via some new analytical skills and Nehair-Poho\v{z}aev identity. In particular, the monotonicity assumption on the nonlinearity is not necessary.