Pointwise estimates of solutions to semilinear elliptic equations and inequalities

TL;DR

This paper derives precise pointwise bounds for positive solutions to a class of semilinear elliptic equations and inequalities involving variable coefficients and sign-changing potentials in Euclidean and Riemannian settings.

Contribution

It provides sharp estimates for solutions to a broad class of elliptic equations with sign-changing potentials, extending previous results to more general operators and geometries.

Findings

Established sharp pointwise bounds for solutions

Extended estimates to weighted Riemannian manifolds

Handled equations with sign-changing potentials

Abstract

We obtain sharp pointwise estimates for positive solutions to the equation $-Lu+Vu^q=f$, where $L$ is an elliptic operator in divergence form, $q\in\mathbb{R}\setminus \{0\}$, $f\geq 0$ and $V$ is a function that may change sign, in a domain $\Omega$ in $\mathbb{R}^{n}$, or in a weighted Riemannian manifold.