= Maximum bound for a monotone reaction term
{title2=$\|u\|_\infty\leq h^{-1}(\|f\|_\infty)$}
Let $h:\mathbb R\to\mathbb R$ be continuous, odd, strictly increasing and onto. Suppose $u\in C^2(\mathbb R^d)$ tends to zero at infinity and satisfies $-\Delta u+h(u)=f$, with bounded $f$. At a positive maximum $m$ of $u$, $-\Delta u\geq0$, so $h(m)\leq f\leq\|f\|_\infty$. Apply the same argument to $-u$ to obtain the displayed bound, using the maximum-point argument of the <weak maximum principle for elliptic operators>. For $h(s)=s+\sin s$, strict monotonicity follows by integrating $h'=1+\cos s$, whose zeros are isolated. Thus $\|u\|_\infty\leq h^{-1}(\|f\|_\infty)\leq\|f\|_\infty+1$. The stronger bound by $\|f\|_\infty$ holds when that norm is at most $\pi$, but need not hold for larger sources.
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