= Failure of the source-size bound for the screened sine-Gordon equation
{title2=$\|u\|_\infty>\|f\|_\infty$}
Take $a=3\pi/2$, $L=10$ and $u(x)=a\exp(-|x|^2/L^2)$ on $\mathbb R^3$. Set $f=-\Delta u+u+\sin u$. Both functions are smooth and belong to <L2 space>, and $u\in H^1$. The strictly increasing reaction $h(s)=s+\sin s$ has $0\leq h(u)\leq h(a)=a-1$, while
$$
|\Delta u|=\frac{a}{L^2}|4q-6|e^{-q}\leq\frac{6a}{L^2},\qquad q=|x|^2/L^2.
$$
Consequently $\|f\|_\infty\leq a-1+6a/L^2<a=\|u\|_\infty$. This $u$ is a minimizer of the <screened sine-Gordon energy>: it solves the <Euler-Lagrange equation> and that energy is convex. Thus even the minimizing and smooth hypotheses do not justify the stronger source-size bound. The appropriate general estimate is <maximum bound for a monotone reaction term>.
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