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I need a maximum principle in unbounded domains: if $u$ is a solution, bounded in $\Omega$, satisfying

$\Delta u+c(x)u=0, \ \ in \ \Omega,$ $c\in L^\infty$, $u\leq0 \ \ in \ \Omega$ $u(x_0)=0, \ \ x_0\in\Omega$ Then $u\equiv0 \ \ in \ \Omega$ Someone know where I can find this statement?

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    Im sorry, $u$ is only nonpositive.2012-11-15

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You can apply the Hopf maximum principle to the operator $Lu = \Delta u - c^{-}u$, where $c^-$ denotes the negative part of $c$: The function $u$ satisfies (since $u\le 0$) $\Delta u - c^- u = -c^+u \ge 0$ The Hopf maximum principle now asserts that $u$ cannot take on an interior maximum in any ball $B_R$ unless $u$ is constant there. In particular, you can take any ball around $x_0$ and see that $u=0$ on $B_R(x_0)$. But this holds for arbitrary $R>0$, so you're done.

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    In the classical book of Trudinger (elliptic equations of second order), you can find a maximum principle in unbounded domains with $c\leq0$, so you dont need analysis locally in balls. Thank you for the demonstration.2012-11-17