Compact embedding from variable-order Sobolev space to $L^{q(x)}(\mathrm{\Omega })$ 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 $W_0^{s(x,y),p(x,y)}(\mathrm{\Omega })$ to the Nakano space $L^{q(x)}(\mathrm{\Omega })$ 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 $p_s^{\ast }(x)$. As an application, we obtain a nontrivial solution of the Choquard equation \begin{equation*} \displaystyle (-Δ)_{p(\cdot,\cdot)}^{s(\cdot,\cdot)}u+|u|^{p(x,x)-2}u=\left(\int_Ω\frac{|u(y)|^{r(y)}}{|x-y|^{\frac{α(x)+α(y)}{2}}}dy\right) |u(x)|^{r(x)-2}u(x)\quad$\text{in }\mathrm{\Omega }$ \end{equation*} with variable upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality under an appropriate boundary condition.