I need to find a specific maximum principle

I need a maximum principle that says:

If $L$ is an elliptic operator and $u$ is a positive function ($u\in C^2(\Omega)$, with $\Omega\subset\mathbb{R}^n$ an unbounded domain) such that $Lu\geq0$ in $\Omega$ and $\limsup_{|x| \to \infty} u\leq0$, then $u\leq0$ in $\Omega$.

If you know a maximum principle like that, please tell me the book or text.

Thank you!

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Questions are supposed to be answered here, not by email. –  Michael Greinecker Oct 1 '12 at 23:12
By the usual maximum principle, $u$ attains its maximum on $B(0,n)$ at some $x_n \in \partial B(0,n)$. Then $u(x_n)$ is an increasing sequence and hence converges to some $M$. Since $|x_n| \to \infty$, we have $M \le 0$. But on the other hand, by definition of $M$ we have $u \le M$ everywhere.