Description

• Title:
  A note on positive solutions of Lichnerowicz equations involving the $\Delta_\lambda$-Laplacian
• Creator:
  Duong, Anh Tuan;
  Nguyen, Thi Quynh
• Description:
  In this paper, we are concerned with the parabolic Lichnerowicz equation involving the $\Delta_\lambda$-Laplacian $$ v_t-\Delta_\lambda v=v^{-p-2}-v^p,\quad v> 0, \quad \mbox{ in }\mathbb R^N\times\mathbb R, $$ where $p> 0$ and $\Delta_\lambda$ is a sub-elliptic operator of the form $$ \Delta_\lambda=\sum_{i=1}^N\partial_{x_i}\big(\lambda_i^2\partial_{x_i}\big). $$ Under some general assumptions of $\lambda_i$ introduced by A.E. Kogoj and E. Lanconelli in Nonlinear Anal. {\bf 75} (2012), no.\ 12, 4637-4649, we shall prove a uniform lower bound of positive solutions of the equation provided that $p> 0$. Moreover, in the case $p> 1$, we shall show that the equation has only the trivial solution $v=1$. As a consequence, when $v$ is independent of the time variable, we obtain the similar results for the elliptic Lichnerowicz equation involving the $\Delta_\lambda$-Laplacian $$ -\Delta_\lambda u=u^{-p-2}-u^p,\quad u> 0,\quad \mbox{in }\mathbb R^N. $$
• Object availability:
  free access
• Rights: https://creativecommons.org/licenses/by-nd/4.0 ;
  Prawa autorskie (c) 2023 Anh Tuan Duong, Thi Quynh Nguyen
• Date:
  2023.09.23
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 62, No 2 (December 2023); 591 - 600;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/46318/36573 ;https://apcz.umk.pl/TMNA/article/view/46318/36574
• Subject:
  Liouville type theorems;
  Lichnerowicz equations;
  parabolic equations;
  elliptic equations;
  uniform lower bound of solutions;
  sub-elliptic operator;
  Primary: 35K40, 35B53, 35J60;
  Secondary: 35B35
• Identifier: https://apcz.umk.pl/TMNA/article/view/46318
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other