Description

• Title:
  Singularly perturbed Neumann problems with potentials
• Creator:
  Pomponio, Alessio
• Description:
  The main purpose of this paper is to study the existence of single-peaked solutions of the Neumann problem $$ \cases -\varepsilon^2 \text{\rm div} \left(J(x)\nabla u\right)+V(x)u=u^p & \text{in }\Omega, \\ \displaystyle \dfrac{\partial u}{\partial \nu}=0 & \text{on }\partial\Omega, \endcases $$ where $\Omega$ is a smooth bounded domain of $\{\mathbb R}^N$, $N\ge 3$, $1< p< (N+2)/(N-2)$ and $J$ and $V$ are positive bounded scalar value potentials. We will show that, for the existence of concentrating solutions, one has to check if at least one between $J$ and $V$ is not constant on $\partial \Omega$. In this case the concentration point is determined by $J$ and $V$ only. In the other case the concentration point is determined by an interplay among the derivatives of $J$ and $V$ calculated on $\partial \Omega$ and the mean curvature $H$ of $\partial \Omega$.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2004.06.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 23, No 2 (June 2004); 301 - 322;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2004.013/9227
• Subject:
  Singularly perturbed Neumann problem;
  presence of potentials;
  concentrating solutions;
  35B25;
  35J20;
  35J55
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2004.013
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other