Denote $$\mathcal{H}=\{u\in H^1(\mathbb{R}^3):\int_{\mathbb{R}^3} V(x)|u(x)|^2\,dx<\infty\},$$ where $V(x)\in L_{loc}^\infty (\mathbb{R}^3)$, $V(x)\geq0$ and $\lim_{|x|\rightarrow \infty} V(x)=\infty$. I want to know that whether the following proposition is right and how to prove it. I already know it's right for $\mathbb{R}^2$.
If $2\leq q<\infty$, then the imbedding $\mathcal{H}\hookrightarrow L^q( \mathbb{R}^3)$ is compact.