Exercise 16, § 3

Let $n\geqslant 1$ and $k\geqslant 1$ be integers. Let $p\geqslant 1$ be a real number. Let U be a bounded open set of $\mathbf{R}^n$.

a) Suppose that $n\not =kp$. Let $q$ be a real number such that $1\leqslant q\leqslant np/(n-kp)$. The inclusion of $\mathscr{D}(U)$ into $L^q(U)$ admits a continuous extension $i$ of $W^{k,p}_0$ (U) into $L^q(U)$.

b) Suppose that $1\leqslant q\leqslant np/(n-kp)$. The linear mapping $i$ is compact ("Rellich-Kondrachov theorem").

c) Suppose that $k\geqslant 1$ and that $q=np/(n-p)$. The linear mapping $i$ is not compact.