Description

• Title:
  The effect of the domain's configuration space on the number of nodal solutions of singularly perturbed elliptic equations
• Creator:
  Bartsch, Thomas;
  Weth, Tobias
• Description:
  We prove a new multiplicity result for nodal solutions of the Dirichlet problem for the singularly perturbed equation $-\varepsilon^2 \Delta u+u =f(u)$ for $\varepsilon> 0$ small on a bounded domain $\Omega\subset{\mathbb R}^N$. The nonlinearity $f$ grows superlinearly and subcritically. We relate the topology of the configuration space $C\Omega=\{(x,y)\in\Omega\times\Omega:x\not=y\}$ of ordered pairs in the domain to the number of solutions with exactly two nodal domains. More precisely, we show that there exist at least $\text{\rm cupl}(C\Omega)+2$ nodal solutions, where $\text{\rm cupl}$ denotes the cuplength of a topological space. We furthermore show that $\text{\rm cupl}(C\Omega)+1$ of these solutions have precisely two nodal domains, and the last one has at most three nodal domains.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2005.09.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 26, No 1 (September 2005); 109 - 133;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2005.027/9280
• Subject:
  Singularly perturbed problem;
  sign-changing solution;
  configuration space;
  Lusternik-Schnirelmn category;
  35J65;
  35B25;
  47J30;
  58E05
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2005.027
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other