One-dimensional symmetry results for semilinear equations and inequalities on half-spaces

2025/09/10 by Beuvin, Nicolas, Farina, Alberto
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.08431

Abstract

We prove new one-dimensional symmetry results for non-negative solutions, possibly unbounded, to the semilinear equation -Δu= f(u) in the upper half-space ℝ^N₊. Some Liouville-type theorems are also proven in the case of differential inequalities in ℝ^N₊, even without imposing any boundary condition. Although subject to dimensional restrictions, our results apply to a broad family of functions f. In particular, they apply to all non-negative f that behaves at least linearly at infinity.

Citations

• Liouville-type theorems for the Lane-Emden equation in the half-space and cones
• Regularity and Symmetry for Semilinear Elliptic Equations in Bounded Domains
• Elliptic partial differential equations from an elementary viewpoint
• Stable solutions to semilinear elliptic equations are smooth up to\n dimension 9
• A new method of proving a priori bounds for superlinear elliptic PDE
• Monotonicity and symmetry of nonnegative solutions to -Δu=f(u) in half-planes and strips
• Qualitative properties and classification of nonnegative solutions to -Δu=f(u) in unbounded domains when f(0)<0
• Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth
• Global and local behavior of positive solutions of nonlinear elliptic equations
• Quasilinear Dirichlet problems driven by positive sources

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