Variable order nonlocal Choquard problem with variable exponents

Abstract

In this article, we study the existence/multiplicity results for the following variable order nonlocal Choquard problem with variable exponents (-Δ)_{p(\cdot)}^{s(\cdot)}u(x)&=λ|u(x)|^{α(x)-2}u(x)+ \left(\DD\int_Ω\frac{F(y,u(y))}{|x-y|^{μ(x,y)}}dy\right)f(x,u(x)), x\in Ω, u(x)&=0, x\in \mathbb R^N\setminusΩ, where $\Omega \subset {\mathbb{R}}^N$ is a smooth and bounded domain, $N\ge 2$, $p,s,\mu$ and $\alpha$ are continuous functions on ${\mathbb{R}}^N\times {\mathbb{R}}^N$ and $f(x,t)$ is Carathédory 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.

21 pages