Description

• Title:
  Mountain pass solutions and an indefinite superlinear elliptic problem on $\mathbb R^{\mathbb N}$
• Creator:
  Du, Yihong;
  Guo, Yuxia
• Description:
  We consider the elliptic problem $$ -\Delta u-\lambda u=a(x) g(u), $$ with $a(x)$ sign-changing and $g(u)$ behaving like $u^p$, $p> 1$. Under suitable conditions on $g(u)$ and $a(x)$, we extend the multiplicity, existence and nonexistence results known to hold for this equation on a bounded domain (with standard homogeneous boundary conditions) to the case that the bounded domain is replaced by the entire space $\mathbb R^N$. More precisely, we show that there exists $\Lambda> 0$ such that this equation on $\mathbb R^N$ has no positive solution for $\lambda> \Lambda$, at least two positive solutions for $\lambda\in (0,\Lambda)$, and at least one positive solution for $\lambda\in (-\infty,0]\cup\{\Lambda\}$. Our approach is based on some descriptions of mountain pass solutions of semilinear elliptic problems on bounded domains obtained by a special version of the mountain pass theorem. These results are of independent interests.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2003.09.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 22, No 1 (September 2003); 69 - 92;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2003.028/9590
• Subject:
  Mountain pass solution;
  Morse index;
  a priori estimates;
  35J20;
  35J60;
  58E05
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2003.028
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other