= Nonnegative truncation of a semilinear source
{title2=$F(s)=f(\max\{s,0\})$}
If $f$ is continuous, bounded and nonnegative on $[0,\infty)$, extend it to a bounded continuous function on the real line by the displayed formula. The <bounded semilinear Dirichlet existence theorem> gives $-\Delta u=F(u)$ with $u\in H_0^1$. Test with the negative part $u_-=\max\{-u,0\}$ to obtain $-\|\nabla u_-\|_2^2=\int F(u)u_-\geq0$. The <Poincaré inequality> gives $u_-=0$, so $u\geq0$ and $F(u)=f(u)$. Thus singularities or uncontrolled growth of the original formula at negative arguments need not prevent a nonnegative <weak solution>.
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