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 Carathedory 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.

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具有可变指数的变阶非局部 Choquard 问题

在本文中，我们研究以下变阶非局部 Choquard 问题的存在/多重性结果，其中变量指数为 (-\Delta)_{p(\cdot)}^{s(\cdot)}u(x)&=\ lambda|u(x)|^{\alpha(x)-2}u(x)+ \left(\DD\int_\Omega\frac{F(y,u(y))}{|xy|^{ \mu(x,y)}}dy\right)f(x,u(x)), x\in \Omega, u(x)&=0, x\in \mathbb R^N\setminus\Omega,其中 $\Omega\subset\mathbb R^N$ 是光滑有界域，$N\geq 2$, $p,s,\mu$ 和 $\alpha$ 是 $\mathbb R^N\ 上的连续函数times\mathbb R^N$ 和 $f(x,t)$ 是 Carathedory 函数。在对 $s,p,\mu,\alpha$ 和 $f(x,t)$ 的适当假设下，首先我们研究适用于变阶分数 Sobolev 空间的变指数的类似 Hardy-Sobolev-Littlewood 型结果和可变指数。然后我们给出上述方程的存在性/多重性结果。