Description

• Title:
  Spectral properties and nodal solutions for second-order, $m$-point, $p$-Laplacian boundary value problems
• Creator:
  Dodds, Niall;
  Rynne, Bryan P.
• Description:
  We consider the $m$-point boundary value problem consisting of the equation $$ -\phi_p (u')'=f(u), \quad \text{on $(0,1)$},\tag 1 $$ together with the boundary conditions $$ u(0) = 0,\quad u(1) = \sum^{m-2}_{i=1}\alpha_i u(\eta_i) ,\tag 2 $$ where $p> 1$, $\phi_p(s) := |s|^{p-1} \text{\rm sgn} s$, $s \in {\mathbb R}$, $m \ge 3$, $\alpha_i , \eta_i \in (0,1)$, for $i=1,\dots,m-2$, and $\sum^{m-2}_{i=1} \alpha_i < 1$. We assume that the function $f \colon {\mathbb R} \to{\mathbb R}$ is continuous, satisfies $sf(s) > 0$ for $s \in {\mathbb R} \setminus \{0\}$, and that $f_0 := \lim_{\xi \rightarrow 0}{f(\xi)}/{\phi_p(\xi)} > 0$. %(we assume that the limit exists and is finite). Closely related to the problem (1), (2), is the spectral problem consisting of the equation $$ -\phi_p (u')' = \la \phi_p(u) , \tag 3 $$ together with the boundary conditions (2). It will be shown that the spectral properties of (2), (3), are similar to those of the standard Sturm-Liouville problem with separated (2-point) boundary conditions (with a minor modification to deal with the multi-point boundary condition). The topological degree of a related operator is also obtained. These spectral and degree theoretic results are then used to prove a Rabinowitz-type global bifurcation theorem for a bifurcation problem related to the problem (1), (2). Finally, we use the global bifurcation theorem to obtain nodal solutions %(that is, sign-changing solutions with a specified number of zeros) of (1), (2), under various conditions on the asymptotic behaviour of $f$.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2008.09.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 32, No 1 (September 2008); 21 - 40;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2008.033/8776
• Subject:
  $m$-point;
  $p$-Laplacian;
  spectral properties;
  nodal solutions;
  34B10;
  34B18;
  34L05;
  34L30
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2008.033
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other