Prove that $\lim\limits_{|x|\rightarrow \infty} u(x) = 0$ for $u \in W^{1,p}(\mathbb{R}^n).$

Question

Let $n < p < \infty$ and $ u \in W^{1,p}(\mathbb{R}^n). $

Prove that $\lim\limits_{|x|\rightarrow \infty} u(x) = 0.$

In the proof I can use that $ C_0^{\infty}(\mathbb{R}^n) $ is dense in $ W^{1,p}(\mathbb{R}^n). $

Can someone help me?

Answer 1

You can see here in these notes by John Hunter. It should be Theorem 3.37 what you are looking for.

Theorem 3.37. Let $n<p<\infty$ and $\alpha =1-n/p$. Then $W^{1,p}(\mathbb{R}^n) \hookrightarrow C_0^{0,\alpha}(\mathbb{R}^n)$ and there is a constant such that $\|f\|_{C^{0,\alpha}} \leq C\|f\|_{W^{1,p}}$ for all $f \in C_c^\infty(\mathbb{R}^n)$.

The proof goes like this. Let $u \in W^{1,p}(\mathbb{R}^n)$, then there exists some $\{u^\epsilon \}_{\epsilon>0}\subset C_c^\infty(\mathbb{R}^n)$ such that $u^\epsilon \to u$ in $W^{1,p}(\mathbb{R}^n)$ as $\epsilon \to 0$. Moreover, we have $|u^\epsilon(x)-u^\epsilon(y)| \leq C|x-y|^{1-n/p}\|D u^\epsilon\|_{L^p}$ for all $x,y$ and hence $|u(x)-u(y)| \leq C|x-y|^{1-n/p}\|D u\|_{L^p}$ for all $x,y$ (by choosing the continuous representation of $u$). Also, $u$ is the uniform limit of a compactly supported function and hence $u(x) \to 0$ as $x\to \pm \infty$.