Description

• Title:
  Existence and concentration of local mountain passes for a nonlinear elliptic field equation in the semi-classical limit
• Creator:
  D'Aprile, Teresa
• Description:
  In this paper we are concerned with the problem of finding solutions for the following nonlinear field equation $$ -\Delta u + V(hx)u-\Delta_{p}u+ W'(u)=0, $$ where $u:\mathbb R^{N}\rightarrow \mathbb R^{N+1}$, $N\geq3$, $p&gt; N$ and $h&gt; 0$. We assume that the potential $V$ is positive and $W$ is an appropriate singular function. In particular we deal with the existence of solutions obtained as critical (not minimum) points for the associated energy functional when $h$ is small enough. Such solutions will eventually exhibit some notable behaviour as $h\rightarrow 0^{+}$. The proof of our results is variational and consists in the introduction of a modified (penalized) energy functional for which mountain pass solutions are studied and soon after are proved to solve our equation for $h$ sufficiently small. This idea is in the spirit of that used in M. Del Pino and P. Felmer [< i> Local mountain passes for semilinear elliptic problems in unbounded domains< /i> , Calc. Var. Partial Differential Equations < b> 4< /b> (1996), 121–137], [< i> Semi-classical states for nonlinear Schrödinger equations< /i> , J. Funct. Anal. < b> 149< /b> (1997), 245–265] and [< i> Multi-peak bound states for nonlinear Schrödinger equations< /i> , Ann. Inst. H. Poincaré Anal. Non Linéaire < b> 15< /b> (1998), 127–149], where "local mountain passes" are found in certain nonlinear Schrödinger equations.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2001.06.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 17, No 2 (June 2001); 239 - 275;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2001.015/9946
• Subject:
  Nonlinear Schrödinger equation;
  existence. concentration;
  topological charge;
  35J20;
  35J60
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2001.015
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other