Description

• Title:
  Nonlinear eigenvalue problems admitting eigenfunctions with known geometric properties
• Creator:
  Heid, Michael;
  Heinz, Hans-Peter
• Description:
  We consider nonlinear eigenvalue problems of the form $$ A_0 y + B(y) y = \lambda y \tag{$*$} $$ in a real Hilbert space $\mathcal H$, where $A_0$ is a semi-bounded self-adjoint operator and, for every $y$ from a certain dense subspace $X$ of $\mathcal H, B(y)$ is a bounded symmetric linear operator. The left hand side is assumed to be the gradient of a functional $\psi \in C^1(x)$, and the associated linear problems $$ A_0 v + B(y) v = \mu v \tag{$**$} $$ are supposed to have discrete spectrum $(y \in X)$. We present a new topological method which permits, under appropriate assumptions, to construct solutions of ($*$) on a sphere $S_R := \{ y \in X \mid \|y\|_{\mathcal H} = R\}$ whose $\psi$-value is the $n$th Ljusternik-Schnirelman level of $\psi |_{S_R}$ and whose corresponding eigenvalue is the $n$th eigenvalue of the associated linear problem ($**$), where $R > 0$ and $n \in \mathbb N$ are given. In applications, the eigenfunctions thus found share any geometric property enjoyed by an $n$-th eigenfunction of a linear problem of the form ($**$). We discuss applications to nonlinear Sturm-Liouville problems, to the nonlinear Hill's equation, to periodic solutions of second-order systems, and to elliptic partial differential equations with radial symmetry.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  1999.03.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 13, No 1 (March 1999); 17 - 51;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.1999.002/10044
• Subject:
  Ljusternik-Schnirelman levels;
  Krasnosel'skiĭ genus;
  nodal properties o solutions;
  nonlinear Sturm-Liouville problems;
  nonlinear Hill's equation;
  semilinear second-order systems;
  periodic solutions;
  Morse index;
  Primary 47H15;
  Secondary 34B15
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.1999.002
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other