This paper is concerned with the existence and multiplicity of solutions for the fractional variable order Choquard type problem $\begin{array}{r}\begin{array}{l}(-\mathrm{\Delta }{)}_{p(\cdot )}^{s(\cdot )}u(x)+(-\mathrm{\Delta }{)}_{q(\cdot )}^{s(\cdot )}u\left(x\right)=\lambda |u\left(x\right){|}^{\text{β}\left(x\right)-2}u\left(x\right)\\ +\left({\int }_{\mathrm{\Omega }}\frac{F(y,u(y\left)\right)}{|x-y{|}^{\mu (x,y)}}\mathrm{d}y\right)f(x,u(x\left)\right)+k\left(x\right)& \mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u\left(x\right)=0& \mathrm{i}\mathrm{n}{\mathbb{R}}^N\setminus \mathrm{\Omega },\end{array}\end{array}$ where $(-\mathrm{\Delta }{)}_{p(\cdot )}^{s(\cdot )}$ and $(-\mathrm{\Delta }{)}_{q(\cdot )}^{s(\cdot )}$ are two fractional Laplace operators with variable order $s(\cdot ):{\mathbb{R}}^{2N}\to (0,1)$ and with different variable exponents $p(\cdot ):{\mathbb{R}}^{2N}\to (1,\mathrm{\infty })$ and $q(\cdot ):{\mathbb{R}}^{2N}\to (1,\mathrm{\infty })$. Here $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded smooth domain with at least $N\ge 2$, λ is a real parameter, β, μ and k are continuous variable parameters, while F is the primitive function of a suitable f. Under some appropriate conditions on β and k, through variational methods, we prove existence and multiplicity of solutions for the above problem.

本文关注分数变阶 Choquard 类型问题的解的存在性和多重性$\begin{array}{r}\begin{array}{l}(-\mathrm{\Delta }{)}_{p(\cdot )}^{s(\cdot )}你(X)+(-\mathrm{\Delta }{)}_{q(\cdot )}^{s(\cdot )}你\left(X\right)=\lambda |你\left(X\right){|}^{\text{β}\left(X\right)-2}你\left(X\right)\\ +\left({\int }_{\mathrm{\Omega }}\frac{F(\mathrm{是的},你(\mathrm{是的}\left)\right)}{|X-\mathrm{是的}{|}^{\mu (X,\mathrm{是的})}}\mathrm{d}\mathrm{是的}\right)F(X,你(X\left)\right)+ķ\left(X\right)& \mathrm{一世}\mathrm{n}\mathrm{\Omega },\\ 你\left(X\right)=0& \mathrm{一世}\mathrm{n}{\mathbb{R}}^{ñ}\setminus \mathrm{\Omega },\end{array}\end{array}$在哪里$(-\mathrm{\Delta }{)}_{p(\cdot )}^{s(\cdot )}$和$(-\mathrm{\Delta }{)}_{q(\cdot )}^{s(\cdot )}$是两个具有可变顺序的小数拉普拉斯算子$s(\cdot )：{\mathbb{R}}^{2ñ}\to (0,1)$并使用不同的变量指数$p(\cdot )：{\mathbb{R}}^{2ñ}\to (1,\mathrm{\infty })$和$q(\cdot )：{\mathbb{R}}^{2ñ}\to (1,\mathrm{\infty })$. 这里$\mathrm{\Omega }\subset {\mathbb{R}}^{ñ}$是一个有界平滑域，至少有$ñ\ge 2$, λ是实参数，β , μ和k是连续变量参数，而F是合适f的原始函数。在β和k的一些适当条件下，通过变分方法，我们证明了上述问题解的存在性和多重性。