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Symmetry of bounded solutions to quasilinear elliptic equations in a half-space

Phuong Le

TL;DR

This work addresses symmetry and monotonicity of bounded positive solutions to the quasilinear elliptic equation $- abla_p u=f(u)$ in the half-space $\,\mathbb{R}^N_+$ with zero boundary data. The authors combine a detailed 1D classification of solutions, variational methods in balls, and weak sweeping principles to prove that, under mild nondegeneracy conditions (notably $f( ho)=0$ with $ ho=\sup u$ and specific behavior near zeros), any bounded solution is one-dimensional and strictly increasing in the normal direction, extending the classical result of Berestycki, Caffarelli and Nirenberg to the $p$-Laplacian. Key contributions include a complete 1D classification of nonnegative solutions, an existence result for positive ball solutions via a variational framework, and a global monotonicity/symmetry conclusion in the half-space for sign-changing nonlinearities. The results provide Rigidity for nonlinear quasilinear elliptic equations and extend foundational symmetry techniques to the $p$-Laplacian setting, with potential implications for related free-boundary and nonlinear diffusion problems.

Abstract

Let $u$ be a bounded positive solution to the problem $-Δ_p u = f(u)$ in $\mathbb{R}^N_+$ with zero Dirichlet boundary condition, where $p>1$ and $f$ is a locally Lipschitz continuous function. Among other things, we show that if $f(\sup_{\mathbb{R}^N_+} u)=0$ and $f$ satisfies some other mild conditions, then $u$ depends only on $x_N$ and monotone increasing in the $x_N$-direction. Our result partially extends a classical result of Berestycki, Caffarelli and Nirenberg in 1993 to the $p$-Laplacian.

Symmetry of bounded solutions to quasilinear elliptic equations in a half-space

TL;DR

This work addresses symmetry and monotonicity of bounded positive solutions to the quasilinear elliptic equation in the half-space with zero boundary data. The authors combine a detailed 1D classification of solutions, variational methods in balls, and weak sweeping principles to prove that, under mild nondegeneracy conditions (notably with and specific behavior near zeros), any bounded solution is one-dimensional and strictly increasing in the normal direction, extending the classical result of Berestycki, Caffarelli and Nirenberg to the -Laplacian. Key contributions include a complete 1D classification of nonnegative solutions, an existence result for positive ball solutions via a variational framework, and a global monotonicity/symmetry conclusion in the half-space for sign-changing nonlinearities. The results provide Rigidity for nonlinear quasilinear elliptic equations and extend foundational symmetry techniques to the -Laplacian setting, with potential implications for related free-boundary and nonlinear diffusion problems.

Abstract

Let be a bounded positive solution to the problem in with zero Dirichlet boundary condition, where and is a locally Lipschitz continuous function. Among other things, we show that if and satisfies some other mild conditions, then depends only on and monotone increasing in the -direction. Our result partially extends a classical result of Berestycki, Caffarelli and Nirenberg in 1993 to the -Laplacian.
Paper Structure (5 sections, 12 theorems, 95 equations)

This paper contains 5 sections, 12 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classification of nonnegative solutions in dimension one
  4. Existence of positive solutions in a ball
  5. Monotonicity and one-dimensional symmetry

Key Result

Theorem 1

Assume $p>1$ and $f:[0,+\infty)\to\mathbb{R}$ is a continuous function which is locally Lipschitz continuous in $[0,+\infty)\setminus Z_f$. Let $u\in C^1(\overline{\mathbb{R}^N_+})$ be a bounded solution to main. Assume that $\rho=\sup_{\mathbb{R}^N_+} u$ verifies $f(\rho)=0$ and $f$ has no zeros in and one of the following conditions hold Then $u$ depends only on $x_N$ and monotone increasing in

Theorems & Definitions (24)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • Theorem 3
  • Remark 4
  • Theorem 4: Strong maximum principle and Hopf's lemma
  • Theorem 5: Strong comparison principle
  • Theorem 6: Boundary point lemma
  • ...and 14 more