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I am reading a paper and have trouble understanding the following step on page 581:

Assume we know that $w: \mathbb{R}^n \to \mathbb{R}$ is smooth and bounded. Furthermore $\Vert w \Vert_{L^\infty}>1$ and $w(x) \to 1$ whenever $\vert x \vert \to \infty.$ Finally the following inequality holds \begin{equation} \Delta w + 2 w(1+\frac{c^2}{4}-w) \geq 0. \end{equation}The author then applys the weak maximum principle to obtain \begin{equation} w \leq 1+\frac{c^2}{4}. \end{equation}

While I found several versions of a "weak maximum principle" (e.g. here), I can't figure out how to apply them to this situation. They all either assume that the domain is bounded and/or there are only derivatives of $w$ (but not $w$ itself) in the PDE.

So my question is:

Can someone give me a hint which version of the weak maximum principle is applicable in this situation and how to apply it?

All help is much appreciated!

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  • $\begingroup$ Hint: you can't have strictly positive Laplacian at a point of local maximum. $\endgroup$ – ˈjuː.zɚ79365 Jun 7 '13 at 8:40
  • $\begingroup$ Aside: not everyone has access to MathSciNet. A direct link to a paper (even if it's behind paywall) is better than a link to a reference to a paper which is itself behind a paywall. The first kind of link at least tells the reader what the paper is, so they can try to find it elsewhere. $\endgroup$ – ˈjuː.zɚ79365 Jun 7 '13 at 10:39
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    $\begingroup$ @user79365: the better solution actually is for the OP to copy the link from the "Make link" button on MathSciNet. That link automatically resolves to the MathSciNet review if a subscription is detected, or also to a bibliographic entry that links to the original paper if not. Compare this link with the one provided by the OP. See also this FAQ item. $\endgroup$ – Willie Wong Jun 7 '13 at 11:02
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Suppose $w(x)>1+c^2/4$ for some $x$. Since the set $\{x: w(x)\ge 1+c^2/4 \}$ is bounded and closed, it is compact. Let $x_0$ be a point of this set where $w$ attains its maximum. Since it is a global maximum of $w$, we have $w(x_0)>1+c^2/4$.

On the other hand, $$\Delta w(x_0)\ge 2w(w-1-c^2/4)>0 \tag1$$ Therefore there is $j\in \{1,\dots,n\}$ such that $$\frac{\partial^2}{\partial x_j^2}w(x_0)>0 \tag2$$ Inequality (2) is incompatible with $w$ (considered as a function of $j$th variable only) having a maximum at $x_0$.

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