Description

• Title:
  Uniqueness of solutions for nonlinear Dirichlet problems with supercritical growth
• Creator:
  Molle, Riccardo;
  Passaseo, Donato
• Description:
  We are concerned with Dirichlet problems of the form $$ \div(|D u|^{p-2}Du)+f(u)=0\quad \mbox{in }\Omega,\qquad u=0\quad \mbox{on }\partial\Omega, $$% where $\Omega$ is a bounded domain of $\mathbb{R}^n$, $n\ge 2$, $1< p< n$ and $f$ is a continuous function with supercritical growth from the viewpoint of the Sobolev embedding. In particular, if $n=2$ and $\gamma\colon [a,b]\to\mathbb{R}^2$ is a smooth curve such that $\gamma(t_1)\neq\gamma(t_2)$ for $t_1\neq t_2$, we prove that, for $\varepsilon> 0$ small enough, there exists a unique solution of the Dirichlet problem in the domain $\Omega=\Omega^\Gamma_\varepsilon=\{(x_1,x_2)\in\mathbb{R}^2 : \dist\big((x_1,x_2),\Gamma\big)< \varepsilon\}$, where $\Gamma=\{\gamma(t) : t\in[a,b]\}$. Moreover, we extend this uniqueness result to the case where $n> 2$ and $\Omega$ is, for example, a domain of the type $$ \Omega=\widetilde\Omega^\Gamma_{\varepsilon,s}=\{(x_1,x_2,y) : (x_1,x_2)\in\Omega^\Gamma_\varepsilon, \ y\in\mathbb{R}^{n-2},\ |y|< s\}. $$
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2021 Topological Methods in Nonlinear Analysis
• Date:
  2021.06.02
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 57, No 2 (June 2021); 535 - 546;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2020.048/29076 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2020.048/29077
• Subject:
  Supercritical Dirichlet problems;
  contractible domains;
  nonexistence of solutions;
  35J20;
  35J60;
  35J65
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2020.048
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other