Estimates of Nonnegative Solutions to Semilinear Elliptic Equations

Let $L$ be a second order uniformly elliptic differential operator in a domain $D$ of $\mathbb{R}^{d}$, $\psi:\mathbb{R}_+\to \mathbb{R}_+$ be a nondecreasing continuous function and let $\xi,g:D\to\mathbb{R}_+$ be locally bounded Borel measurable functions. Under appropriate conditions, we determine a function $\varphi$ with values in $]0,1]$ such that for every nonnegative solution to inequality $-Lu+\xi\psi(u) \geq g$ in $D$ and for every $x\in D$, $$u(x)\geq p(x)\,\varphi\left(\frac{G_D(\xi\psi(p))(x)}{p(x)}\right)$$ where $p=G_Dg$ is the Green function of $g$. The function $\varphi$ is completely determined by $\psi$ and does not depend on $L,D,\xi$ or $g$.