Compact Imbedding on weighted Sobolev Spaces

I am trying to solve the simple Eigenvalue problem

$$-\Delta{u}=\lambda u, \; u = 0$$ on $$\partial{\Omega}$$

in Sobolev spaces on a smooth exterior (and thus unbounded) domain $$\Omega \subset \mathbb{R}^{n}$$. Unfortunately, the Rellich-Kondrachov compactness theorem cannot be used on $$\Omega$$.

I read something online about using weighted Sobolev Spaces, but didn´t find any concrete references and now tried to do it on my own.

I am trying to prove the compactness of the embedding of $$H^{1}_{0}(\Omega,\omega)$$ in $$L^{2}(\Omega,\omega)$$, where $$H^{1}_{0}(\Omega,\omega)$$ is the completion of $$C^{\infty}_{c}(\Omega)$$ in regards to the norm $$||u||^{2}:=\int_{\Omega}|u|^{2}\omega + |Du|^{2}$$, where $$\omega(x)=1/|x|^{2+\delta}$$ for some small $$\delta > 0$$.

My idea: Let $$(u_{n}) \subset H^{1}_{0}(\Omega,\omega)$$ be a bounded set. Using the Hardy inequality i get $$\int_{\Omega/B_{R}(0)}|u_{n}|^{2}\omega \le 1/|R|^{\delta} \cdot C \cdot \int_{\Omega/B_{R}(0)}|Du_{n}|^{2}$$, which can be made $$< \epsilon$$ for sufficiently large $$R$$.

On $$B_{R}(0)\cap \Omega$$ I use the classic Rellich-Kondrachov theorem to find a finite number of functions in $$L^{2}(\Omega\cap B_{R}(0),\omega)$$ which cover the $$u_{n}$$ up to $$\epsilon$$. Thus, the $$u_{n}$$ are totally bounded.

Does this make sense?

• Do you assume that $0\in \Omega$ so that $L^2(\Omega\cap B_R)=L^2(\Omega\cap B_R,w)$ with equivalent norms? Otherwise I don't see how you use Rellich-Kondrachov for the bounded part. The rest seems fine to me. – MaoWao Feb 11 at 15:19
• Thanks! I will adjust $\omega$ slightly to make it work in the general case of $0 \notin \Omega$. One more question: Since the dual $(H^{1}_{0}(\Omega,\omega))´$ is way smaller than the weighted space itself, I am having trouble applying the methods one usually uses with compact operators. Could you briefly explain how I use the spectral theorem in this case? – Falc14 Feb 12 at 10:52