# Variable order nonlocal Choquard problem with variable exponents

## TL;DR

This paper investigates the existence and multiplicity of solutions for a complex variable order nonlocal Choquard equation involving variable exponents, extending fractional Sobolev space theory and establishing new Hardy-Sobolev-Littlewood-type results.

## Contribution

It introduces new existence and multiplicity results for a variable order nonlocal Choquard problem with variable exponents, expanding the theoretical framework for fractional Sobolev spaces.

## Key findings

- Established Hardy-Sobolev-Littlewood-type inequalities for variable exponents.
- Proved existence of solutions under certain conditions.
- Demonstrated multiplicity of solutions for the problem.

## Abstract

In this article, we study the existence/multiplicity results for the following variable order nonlocal Choquard problem with variable exponents (-\Delta)_{p(\cdot)}^{s(\cdot)}u(x)&=\lambda|u(x)|^{\alpha(x)-2}u(x)+ \left(\DD\int_\Omega\frac{F(y,u(y))}{|x-y|^{\mu(x,y)}}dy\right)f(x,u(x)), x\in \Omega, u(x)&=0, x\in \mathbb R^N\setminus\Omega, where $\Omega\subset\mathbb R^N$ is a smooth and bounded domain, $N\geq 2$, $p,s,\mu$ and $\alpha$ are continuous functions on $\mathbb R^N\times\mathbb R^N$ and $f(x,t)$ is Carath\'edory function. Under suitable assumption on $s,p,\mu,\alpha$ and $f(x,t)$, first we study the analogous Hardy-Sobolev-Littlewood-type result for variable exponents suitable for the fractional Sobolev space with variable order and variable exponents. Then we give the existence/multiplicity results for the above equation.