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

2025/09/10 by Nicolas Beuvin, Alberto Farina, Beuvin, Nicolas +1
Computer Science · Mathematics · #Analysis of PDEs (math.AP) #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2509.08431

openalex publication_date 2025/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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

Related

• Symmetry results in the half space for a semi-linear fractional Laplace equation through a one-dimensional analysis
• One-dimensional symmetry of positive bounded solutions to the nonlinear Schrödinger equation in the half-space
• A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane
• Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian
• One-dimensional symmetry of positive bounded solutions to the subcubic and cubic nonlinear Schrödinger equation in the half-space in dimensions N=4,5