Monotonicity and Liouville-type theorems for semilinear elliptic problems in the half space

2025/10/02 by Sciunzi, Berardino, Vuono, Domenico
#35B06 #35B51 #35J61 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.01865

Abstract

We consider classical solutions to -Δu = f(u) in half-spaces, under homogeneous Dirichlet boundary conditions. We prove that any positive solution is strictly monotone increasing in the direction orthogonal to the boundary, provided that it is directionally bounded on finite strips. As a corollary, we deduce a new Liouville-type theorem for the Lane-Emden equation.

Citations

• Monotonicity results in half spaces for quasilinear elliptic equations involving a singular term
• Liouville-type theorems for the Lane-Emden equation in the half-space and cones
• Qualitative properties and classification of nonnegative solutions to -Δu=f(u) in unbounded domains when f(0)<0
• Monotonicity of solutions to quasilinear problems with a first-order term in half-spaces
• Monotonicity of solutions of quasilinear degenerate elliptic equation in half-spaces
• A symmetry problem in potential theory

Related

• Monotonicity of solutions to quasilinear problems with a first-order term in half-spaces
• Liouville-type theorems for unbounded solutions of elliptic equations in half-spaces
• Monotonicity results in half spaces for quasilinear elliptic equations involving a singular term
• Monotonicity of solutions of quasilinear degenerate elliptic equation in half-spaces
• A Liouville theorem for the Lane-Emden system in the half-space