Let $U= (0,1)\times (0,1)$. Consider the weighted Sobolev spaces $H_1$ with the norms $$ \|u\|_{H_1}^2 = \int_0^1 (\int_0^1 x\,|u(x,y)|^2\,dx) \,dy$$ Let $H_2$ be the weighted Sobolev space with the norm $$ \|u\|_{H_2}^2= \int_0^1 (\int_0^1 x^2(|\partial_x u(x,y)|^2+|\partial_y u(x,y)|^2)\,dx) \,dy+ \int_0^1 (\int_0^1 x\,|u(x,y)|^2\,dx) \,dy$$ subject to the additional constraint that $u(x,1)=u(x,0)=u(1,y)=0$ for all $x,y \in (0,1)$.
Is the embedding $H_2 \subset H_1$ compact?
Assume $\|u\|_{H^2} \leq 1$ and by H"older $$|u(x,y)|\leq \int_x^1 |u_x(t,y)|\, dt \leq \left (\int_x^1 t^2 u_x^2(t,y)\, dt\right )^{\frac 12}\left (\int_x^1 \frac{1}{t^2}\right )^{\frac 12} \leq \frac{C}{\sqrt x} $$ $C= \sqrt 2$, for example. Given $\delta$ we have, $$ \int_0^{\delta^2} \int_0^1 xu^2(x,y)dx dy \leq C\delta^2. $$ Since in $Q_\delta=[\delta^2,1]\times [0,1]$ the norms in $H_1, H_2$ are equivalent to those of $L^2, W^{1,2}$, by the compactness of Sobolev embeddng we can cover the unit ball of $H_2(Q_\delta)$ with a finite number of $\delta$-balls in $H_1(Q_\delta)$ and the same balls give a finite covering (with a slightly changed $\delta$) of the unit ball of $H_2$.
