Estimates of Nonnegative Solutions to Semilinear Elliptic Equations

Abstract

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 $\phi$ with values in $]0,1]$ such that for every nonnegative solution to inequality $-Lu+\xi \psi (u)\ge g$ in $D$ and for every $x\in D$, $$u(x)\ge p(x)\phi \left(\frac{G_D(\xi \psi (p))(x)}{p(x)}\right)$$ where $p=G_{D}g$ is the Green function of $g$. The function $\phi$ is completely determined by $\psi$ and does not depend on $L,D,\xi$ or $g$.

26 pages