Description

• Title:
  Eigenvalue criteria for existence of positive solutions of second-order, multi-point, $p$-Laplacian boundary value problems
• Creator:
  Rynne, Bryan P.
• Description:
  In this paper we consider the existence and uniqueness of positive solutions of the multi-point boundary value problem $$ \gather - (\phi_p(u')' + (a + g(x,u,u'))\phi_p(u) = 0 , \quad\text{a.e. on $(-1,1)$}, \tag{1} \\ u(\pm 1) = \sum^{m^\pm}_{i=1}\al^\pm_i u(\eta^\pm_i) , \tag{2} \endgather $$ where $p> 1$, $\phi_p(s) := |s|^{p-2} s$, $s \in \mathbb R$, $m^\pm \ge 1$ are integers, and $$ \eta_i^\pm \in (-1,1),\quad \al_i^\pm > 0,\quad i = 1,\dots,m^\pm, \quad \sum^{m^\pm}_{i=1} \al_i^\pm < 1 . $$ Also, $a \in L^1(-1,1),$ and $g \colon [-1,1] \X \mathbb R^2 \to \mathbb R$ is Carathéodory, with $$ g(x,0,0) = 0, \quad x \in [-1,1]. \tag{3} $$ Our criteria for existence of positive solutions of (1), (2) will be expressed in terms of the asymptotic behaviour of $g(x,s,t)$, as $s \to \infty$, and the principal eigenvalues of the multi-point boundary value problem consisting of the equation $$ -\phi_p (u')' + a \phi_p (u) = \la \phi_p (u) , \quad \text{on $(-1,1)$}, \tag{4} $$
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2010.04.23
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 36, No 2 (December 2010); 311 - 326;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2010.038/8458
• Subject:
  Positive solutions of nonlinear boundary value problems;
  principal eigenvalues;
  Primary 34B10;
  Secondary 34B18
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2010.038
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other