Multiplicity and uniform estimate for a class of variable order fractional $p(x)$-Laplacian problems with concave-convex nonlinearities

TL;DR

This paper investigates the existence and multiplicity of solutions for a complex variable order fractional p(x)-Laplacian problem with nonlinearities, extending Hardy-Sobolev-Littlewood theory to variable exponents and fractional Sobolev spaces.

Contribution

It introduces new existence and multiplicity results for a variable order nonlocal Choquard problem with variable exponents, incorporating Hardy-Sobolev-Littlewood-type inequalities.

Findings

Established Hardy-Sobolev-Littlewood inequalities for variable exponents.

Proved existence of solutions under certain conditions.

Demonstrated multiple 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 \begin{equation*} \begin{array}{rl} (-\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)),\\ &~\hspace{6cm} x\in \Omega, \\ u(x)&=0 ,\hspace{20mm} x\in \Omega^c:=\mathbb R^N\setminus\Omega, \end{array} \end{equation*} where $\Om\subset\mathbb R^N$ 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 continuous function with $F(x,t):=\displaystyle\int_{0}^{t} f(x,s)ds$. 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.