Let $\Omega \subset R^n$ be a bounded domain with smooth boundary. Let $u \in C^{2}(\Omega) \cap C(\overline{\Omega})$ a function that satisfies $\Delta u = u^3$ in $\Omega$ and $u = 0$ on $\partial \Omega $. Prove that $u$ is the null function.

I have no idea about a first step. Someone could help me ?

Thanks in advance

  • $\begingroup$ This is a guess and might be totally wrong but it feels like a "obtains maximum on boundary" type thing. $\endgroup$ – user223391 Sep 27 '15 at 19:46
  • $\begingroup$ Did you see the solution below ? $\endgroup$ – Svetoslav Mar 18 '16 at 10:17

Multiply both sides by $u$ and integrate over $\Omega$. Then use integration by parts and the fact that $u=0$ on $\partial \Omega$ to get $$\int\limits_{\Omega}{|\nabla u|^2dx}+\int\limits_{\Omega}{u^4dx}=0$$ From here it is clear that $u$ must be zero everywhere in $\Omega$


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