Suppose $ u \in W^{1,p}(R^n) $ , and u is continuous. Then is it true that $ |u(x)| \to 0 $ as $|x| \to \infty $. It is true that, if $$ u \in W^{1,p}(R^n) \text{ then }||u||_{W^{1,p}(R^n\setminus B(0,R))} \to 0 \text{ as } R \to \infty.$$ How to show that $|u(x)| \to 0$ as $|x| \to \infty$.
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You cant. Think about a bunch of bumps headed off to $\infty$. – Umberto P. Sep 04 '22 at 12:08
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@UmbertoP. Will this type of functions belong to $L^p(\mathbb{R}^N)$? – Arun Sep 04 '22 at 12:43
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Yes, and so will their derivatives if you define everything properly. – Umberto P. Sep 04 '22 at 14:30