Compact embedding from variable-order Sobolev space to $L^{q(x)}(\Omega)$ 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 $W^{s(x,y),p(x,y)}_0 (\Omega)$ to the Nakano space $L^{q(x)}(\Omega)$ with a critical exponent $q(x)$ satisfying some conditions. It is noteworthy that the embedding can be compact even when $q(x)$ reaches the critical Sobolev exponent $p_s^*(x)$. As an application, we obtain a nontrivial solution of the Choquard equation \begin{equation*} \displaystyle (-\Delta)_{p(\cdot,\cdot)}^{s(\cdot,\cdot)}u+|u|^{p(x,x)-2}u=\left(\int_{\Omega}\frac{|u(y)|^{r(y)}}{|x-y|^{\frac{\alpha(x)+\alpha(y)}{2}}}dy\right) |u(x)|^{r(x)-2}u(x)\quad\text{in $\Omega$} \end{equation*} with variable upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality under an appropriate boundary condition.