Sufficient conditions for the compact immersion of $H_0^1(\Omega)$ into $L^2(\Omega)$

Question

Is it true that if $\Omega$ is an open set of $\mathbb{R}^d$ (not necessarily a bounded one) on which the following functional admits a minimum point $$ \int_{\Omega} \lvert \nabla u \rvert ^2 \quad \text{with} \quad u \in H_0^1(\Omega),\, \int_{\Omega} u^2 dx=1;$$ then the inclusion of $H_0^1(\Omega)$ in $L^2(\Omega)$ is compact? If it is not true could someone please give me a counterexample?