Compact embedding from variable-order Sobolev space to Lq(x)(Ω) and its application to Choquard equation with variable order and variable critical exponent

In this paper, we prove the compact embedding from the variable-order Sobolev space $W_0^{s(x,y),p(x,y)}\left(\Omega \right)$ to the Nakano space $L^{q\left(x\right)}\left(\Omega \right)$ with a critical exponent $q\left(x\right)$ satisfying some conditions. It is noteworthy that the embedding can be compact even when $q\left(x\right)$ reaches the critical Sobolev exponent $p_s^{⁎}\left(x\right)$. As an application, we obtain a nontrivial solution of the Choquard equation${(-\Delta )}_{p(\cdot ,\cdot )}^{s(\cdot ,\cdot )}u+|u{|}^{p(x,x)-2}u=\left(\underset{\Omega }{\int }\frac{\left|u\right(y){|}^{r\left(y\right)}}{|x-y{|}^{\frac{\alpha \left(x\right)+\alpha \left(y\right)}{2}}}dy\right)|u\left(x\right){|}^{r\left(x\right)-2}u\left(x\right)\text{in }\Omega$ with variable upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality under an appropriate boundary condition.