Description

• Title:
  On a second order boundary value problem with singular nonlinearity
• Creator:
  Benci, Vieri;
  Micheletti, Anna Maria;
  Shteto, Edlira
• Description:
  In this paper we investigate in a variational setting, the elliptic boundary value problem $-\Delta u={\textt{sign}u}/{|u|^{\alpha+1}}$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is an open connected bounded subset of ${\mathbb R}^N$, and $\alpha> 0$. For the positive solution, which is checked as a minimum point of the formally associated functional $$ E(u)=\frac 12\int_\Omega|\nabla u|^2+\frac{1}{\alpha} \int_\Omega \frac1{|u|^\alpha}, $$ we prove dependence on the domain $\Omega$. Moreover, an approximative functional $E_\varepsilon$ is introduced, and an upper bound for the sequence of mountain pass points $u_\varepsilon$ of $E_\varepsilon$, as $\varepsilon\to 0$, is given. For the onedimensional case, all sign-changing solutions of $-u''={\text{sign}u}/{|u|^{\alpha+1}}$ are characterized by their nodal set as the mountain pass point and $n$-saddle points ($n> 1$) of the functional $E$.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2006.03.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 27, No 1 (March 2006); 1 - 28;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2006.001/8861
• Subject:
  Variational methods;
  elliptic problems;
  singular nonlinearity;
  35Q55
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2006.001
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other