Description

• Title:
  Radial symmetry of $n$-mode positive solutions for semilinear elliptic equations in a disc and its application to the Hénon equation
• Creator:
  Shioji, Naoki;
  Watanabe, Kohtaro
• Description:
  Let $f \in C((0,1)\times (0,\infty),\mathbb{R})$ and $n \in \mathbb{N}$ with $n \geq 2$ such that for each $u \in (0,\infty)$, $r\mapsto r^{2-2n}f(r,u)\colon (0,1)\rightarrow \mathbb{R}$ is nonincreasing and let $D=\{x=(x_1,x_2)\in\mathbb{R}^2: |x|< 1\}$. We show that each positive solution of $$ \Delta u + f(|x|,u) =0 \quad\text{in $D$,} \qquad u=0 \quad\text{on $\partial D$} $$ which satisfies $u(r,\theta)= u(r,\theta+2\pi/n)$ by the polar coordinates is radially symmetric and $u_r(|x|)< 0$ for each $r=|x| \in (0,1)$. We apply our result to the Hénon equation.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2016.04.12
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 43, No 1 (March 2014); 269 - 285;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2014.016/8130
• Subject:
  Elliptic equations;
  positive solutions;
  radial symmetry;
  Riemannian isometric;
  the Hénon equation;
  35B06;
  35B09;
  35J15
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2014.016
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other