Maximum bound for a monotone reaction term ($\Vert u{\Vert }_{\infty }\le h^{-1}(\Vert f{\Vert }_{\infty })$)

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 $u$, $-\Delta u\ge 0$, so $h(m)\le f\le \Vert f{\Vert }_{\infty }$. 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 $h^{\prime }=1+\cos s$, whose zeros are isolated. Thus $\Vert u{\Vert }_{\infty }\le h^{-1}(\Vert f{\Vert }_{\infty })\le \Vert f{\Vert }_{\infty }+1$. The stronger bound by $\Vert f{\Vert }_{\infty }$ holds when that norm is at most $\pi$, but need not hold for larger sources.