Description

• Title:
  A singular perturbed problem with critical Sobolev exponent
• Creator:
  Chen, Mengyao;
  Li, Qi
• Description:
  This paper deals with the following nonlinear elliptic problem \begin{equation}\label{eq0.1} -\varepsilon^2\Delta u+\omega V(x)u=u^{p}+u^{2^{*}-1},\quad u> 0\quad\text{in}\ \R^N, \end{equation} where $\omega\in\R^{+}$, $N\geq 3$, $p\in (1,2^{*}-1)$ with $2^{*}={2N}/({N-2})$, $\varepsilon> 0$ is a small parameter and $V(x)$ is a given function. Under suitable assumptions, we prove that problem (\ref{eq0.1}) has multi-peak solutions by the Lyapunov-Schmidt reduction method for sufficiently small $\varepsilon$, which concentrate at local minimum points of potential function $V(x)$. Moreover, we show the local uniqueness of positive multi-peak solutions by using the local Pohozaev identity.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2021 Topological Methods in Nonlinear Analysis;
  https://creativecommons.org/licenses/by-nd/4.0
• Date:
  2021.09.12
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 58, No 1 (September 2021); 181 - 207;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/35211/29668 ;https://apcz.umk.pl/TMNA/article/view/35211/29669
• Subject:
  Multi-peak solutions;
  Lyapunov-Schmidt reduction;
  local uniqueness;
  local Pohozaev identity;
  Primary;
  35B25;
  Secondary;
  35A01;
  35A02;
  35J60
• Identifier: https://apcz.umk.pl/TMNA/article/view/35211
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other