Show that $u \mapsto \int_{\Omega} \nabla u_0 \nabla u $ is a compact operator?
where $\Omega$ is a bounded subset of $R^n,$ $\Omega$ satisfies the codintion of the sobolev embedding theorem, and $u_0,u \in H_0^1 \left( \Omega \right).$
Could anyone help me to give a solution for the problem! Thank you!