If $u\in W^{1,p}(\mathbb{R}^N)$ is a solution of a pde problem , what it means $$\lim_{|x|\rightarrow+\infty}u(x)=0$$
2026-04-04 14:25:12.1775312712
What it means $\lim_{|x|\rightarrow+\infty}u(x)=0$?
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One posible interpretation is the following: for every $\epsilon > 0$, there exist a compact $K_\epsilon \subset \Bbb R ^N$ such that $\text{ess sup} \left| u \big|_{\Bbb R^N \setminus K_\epsilon} \right| < \epsilon$, where $\text{ess sup}$ is the essential supremum.