Bounded semilinear Dirichlet existence theorem ($-\Delta u=f(u),u\in H_0^1(\Omega ),f\in C_b(\mathbb{R})$)

For a smooth bounded domain and a bounded continuous scalar function $f$, this problem has a weak solution. Let $S$ be the zero-Dirichlet inverse of $-\Delta$, defined by the Lax-Milgram theorem. The Poincaré inequality gives $\Vert Sh{\Vert }_2\le C_P^2\Vert h{\Vert }_2$. The Rellich-Kondrachov compactness theorem makes $S:L^2\to L^2$ compact. The Nemytskii operator $N_f$ is continuous and has bounded image, so $T=S\circ N_f$ is a nonlinear compact operator mapping a sufficiently large closed $L^2$ ball into itself. The Schauder fixed-point theorem gives a fixed point, necessarily in $H_0^1$, and its defining identity is the desired weak formulation. This proves existence; uniqueness requires additional assumptions.

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 $-\Vert \nabla u_{-}{\Vert }_2^2=\int F(u)u_{-}\ge 0$. The Poincaré inequality gives $u_{-}=0$, so $u\ge 0$ and $F(u)=f(u)$. Thus singularities or uncontrolled growth of the original formula at negative arguments need not prevent a nonnegative weak solution.