# Multiple and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent

## TL;DR

This paper investigates the existence and multiplicity of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent, linking solutions to the topology of the potential's minimum set and their concentration behavior.

## Contribution

It establishes the existence of ground state solutions and relates their number and concentration to the topology of the potential's minimum set.

## Key findings

- Existence of nontrivial nonnegative ground state solutions.
- Number of solutions linked to the topology of the potential's minimum set.
- Solutions exhibit concentration behavior around minimum points.

## Abstract

In present paper, we study the fractional Choquard equation $$\varepsilon^{2s}(-\Delta)^s u+V(x)u=\varepsilon^{\mu-N}(\frac{1}{|x|^\mu}\ast F(u))f(u)+|u|^{2^\ast_s-2}u$$ where $\varepsilon>0$ is a parameter, $s\in(0,1),$ $N>2s,$ $2^*_s=\frac{2N}{N-2s}$ and $0<\mu<\min\{2s,N-2s\}$. Under suitable assumption on $V$ and $f$, we prove this problem has a nontrivial nonnegative ground state solution. Moreover, we relate the number of nontrivial nonnegative solutions with the topology of the set where the potential attains its minimum values and their's concentration behavior.