Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $\Omega\in\mathbf{R}^N$ an unbounded domain and $u\in C^2(\Omega)\cap C(\overline\Omega)$, $u>0$ such that $$\Delta u + f(u)=0, \ \ \ \mbox{em} \ \ \Omega,$$ where $f$ is a bounded lipschitz continuous function. Then $u$ is bounded.

I don't know where I can find this result, and I believe that this assumptions implies $\nabla u$ is bounded too. Someone can help me?

share|cite|improve this question
up vote 1 down vote accepted

This appears to be a counterexample: $\Omega=\{(x,y)\in \mathbb R^2: x>1\}$; $u(x,y)=x$; $f\equiv 0$.

share|cite|improve this answer
Yes... you give the counter example, thanks. But, you know that this result is true to the first derivate? $|\nabla u|$ is bounded in this situation? – José Carlos Dec 15 '12 at 20:37
@JoséCarlos On the same domain, $u(x,y)=x^2$ and $f\equiv 2$ give an example with unbounded gradient. – user53153 Dec 15 '12 at 20:51
Pavel thank you for the counter examples! Im going to search these estimates at least in bounded domains. – José Carlos Dec 17 '12 at 1:08
@JoséCarlos I think the key term for that is "Alexandrov-Bakelman-Pucci". – user53153 Dec 17 '12 at 1:37
Thank you very much! – José Carlos Dec 18 '12 at 1:04

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.