Log-concavity and anti-maximum principles for semilinear and linear elliptic equations
François Hamel, Nikolai Nadirashvili
TL;DR
The paper studies positive solutions of semilinear elliptic equations in bounded domains with Dirichlet boundary data, focusing on perturbations near the linear eigenproblem and on convexity-driven concavity properties. It combines Schauder fixed point methods, maximum and quantified anti-maximum principles, and a priori estimates to establish existence and asymptotic behavior as perturbation parameters vanish. The main results show existence when the perturbation signs match and the magnitudes are small, convergence of solutions to a positive multiple of the principal eigenfunction $\varphi_1$, and log-concavity of solutions in smooth, strictly convex domains, with a proof strategy that leverages convergence to $\varphi_1$ and the known log-concavity of the eigenfunction. These findings deepen understanding of near-linear elliptic problems and provide sharper convexity/concavity information for nonlinear perturbations.
Abstract
This paper is concerned with existence and qualitative properties of positive solutions of semilinear elliptic equations in bounded domains with Dirichlet boundary conditions. We show the existence of positive solutions in the vicinity of the linear equation and the log-concavity of the solutions when the domain is strictly convex. We also review the standard results on the log-concavity or the more general quasi-concavity of solutions of elliptic equations. The existence and other convergence results especially rely on the maximum principle, on a quantified version of the anti-maximum principle, on the Schauder fixed point theorem, and on some a priori estimates.