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Uniqueness of the critical points of solutions to two kinds of semilinear elliptic equations in higher dimensional domains

Haiyun Deng, Jingwen Ji, Feida Jiang, Jiabin Yin

TL;DR

The paper addresses Conjecture A for positive stable solutions of the semilinear Dirichlet problem $-\Delta u=f(x,u)$ in bounded domains $\Omega\subset \mathbb{R}^n$ with $n\ge 3$, focusing on domains that are simple rotationally symmetric about the $x_n$-axis. It develops a continuity method together with maximum principles (Serrin-type comparisons, BV theorem, and corner Hopf lemmas) and a Morse/stability framework to propagate symmetry and monotonicity as the domain deforms from a ball to $\Omega$. The main contributions show that, for both cases where $f(x,u)$ is convex in $u$ (with additional symmetry/monotonicity in the $x$-variables) and where $f(u)$ is convex in $u$, positive stable solutions are axisymmetric about the $x_n$-axis, strictly decreasing in the radial direction, and possess a unique nondegenerate critical point in $\Omega$, thereby affirming Conjecture A on these domains. These results extend uniqueness of critical points to higher dimensions under milder domain geometry and suggest applicability to a broader class of uniformly elliptic equations beyond the classical projection methods.

Abstract

In this paper, we provide an affirmative answer to the {\it conjecture A} for bounded simple rotationally symmetric domains $Ω\subset \mathbb{R}^n(n\geq 3)$ along $x_n$ axis. Precisely, we use a new simple argument to study the symmetry of positive stable solutions for two kinds of semilinear elliptic equations. To do this, when $f(\cdot,s)$ is convex with respect to $s$, we show that the positivity of the first eigenvalue of the corresponding linearized operator in somehow symmetric domains is a sufficient condition for the symmetry of $u$. Moreover, we prove the uniqueness of critical points of a positive stable solution to semilinear elliptic equation $-\triangle u=f(\cdot,u)$ with zero Dirichlet boundary condition for simple rotationally symmetric domains in $\mathbb{R}^n$ by continuity method and a variety of maximum principles.

Uniqueness of the critical points of solutions to two kinds of semilinear elliptic equations in higher dimensional domains

TL;DR

The paper addresses Conjecture A for positive stable solutions of the semilinear Dirichlet problem in bounded domains with , focusing on domains that are simple rotationally symmetric about the -axis. It develops a continuity method together with maximum principles (Serrin-type comparisons, BV theorem, and corner Hopf lemmas) and a Morse/stability framework to propagate symmetry and monotonicity as the domain deforms from a ball to . The main contributions show that, for both cases where is convex in (with additional symmetry/monotonicity in the -variables) and where is convex in , positive stable solutions are axisymmetric about the -axis, strictly decreasing in the radial direction, and possess a unique nondegenerate critical point in , thereby affirming Conjecture A on these domains. These results extend uniqueness of critical points to higher dimensions under milder domain geometry and suggest applicability to a broader class of uniformly elliptic equations beyond the classical projection methods.

Abstract

In this paper, we provide an affirmative answer to the {\it conjecture A} for bounded simple rotationally symmetric domains along axis. Precisely, we use a new simple argument to study the symmetry of positive stable solutions for two kinds of semilinear elliptic equations. To do this, when is convex with respect to , we show that the positivity of the first eigenvalue of the corresponding linearized operator in somehow symmetric domains is a sufficient condition for the symmetry of . Moreover, we prove the uniqueness of critical points of a positive stable solution to semilinear elliptic equation with zero Dirichlet boundary condition for simple rotationally symmetric domains in by continuity method and a variety of maximum principles.
Paper Structure (4 sections, 10 theorems, 70 equations)

This paper contains 4 sections, 10 theorems, 70 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Proof of Theorem \ref{['thm1.1']}
  4. Proof of Theorem \ref{['thm1.2']}

Key Result

Theorem 1.1

Let $\Omega\subset \mathbb{R}^n(n\geq 3)$ be a bounded simple rotationally symmetric domain along $x_n$ axis. Let $u$ be a positive stable solution to the following equation where $f(x,u)\in C^{1,1}(\Omega,\mathbb{R})$ is a convex function with respect to $u$, $f(x^\prime,x_n,u)=f(x^\prime,-x_n,u)$, $f(x^\prime,x_n,u)=f(|x^\prime|,x_n,u)$, $x^\prime=(x_1,\cdots,x_{n-1})$ and $f$ is decreasing in

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • Lemma 2.2
  • ...and 13 more