Differential and Integral Equations

Half-eigenvalues of self-adjoint, $2m$th-order differential operators and semilinear problems with jumping nonlinearities

Bryan P. Rynne

Abstract

We consider semilinear boundary value problems of the form \begin{equation} L u(x) = f(x,u(x)) + h(x), \quad x \in (0,\pi), \tag*{(1)} \end{equation} where $L$ is a $2m$th-order, self-adjoint, disconjugate ordinary differential operator on $[0,\pi]$, together with appropriate boundary conditions at $0$ and $\pi$, while $f : [0,\pi] {\times} \mathbb R {\rightarrow} \mathbb R$ is a Carath{\'e}odory function and $h \in L^2(0,\pi)$. We assume that the limits $$a(x) := \lim_{\xi {\rightarrow} \infty} f(x,\xi)/\xi, \quad b(x) := \lim_{\xi {\rightarrow} -\infty} f(x,\xi)/\xi,$$ exist for almost every $x \in [0,\pi]$ and $a,\,b \in L^{\infty}(0,\pi)$, but $a \ne b$. In this case the nonlinearity $f$ is termed {\em jumping}. Closely related to (1) is the "limiting" boundary value problem \begin{equation} L u = a u^+ - b u^- +{\lambda} u + h, \tag*{(2)} \end{equation} where $u^{\pm}(x) = \max\{\pm u(x),0\}$ for $x \in [0,\pi]$, and ${\lambda}$ is a real parameter. Values of ${\lambda}$ for which (2) (with $h=0$) has a nontrivial solution $u$ will be called {\em half-eigenvalues} of $(L;a,b)$. In this paper we show that a sequence of half-eigenvalues exists, with certain properties, and we prove various results regarding the existence and multiplicity of solutions of both (1) and (2). These result depend strongly on the location of the half-eigenvalues relative to the point ${\lambda}=0$. Some geometric properties of the Fučík spectrum of $L$ are also briefly discussed.

Article information

Source
Differential Integral Equations, Volume 14, Number 9 (2001), 1129-1152.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356124311

Mathematical Reviews number (MathSciNet)
MR1852875

Zentralblatt MATH identifier
1028.34075

Citation

Rynne, Bryan P. Half-eigenvalues of self-adjoint, $2m$th-order differential operators and semilinear problems with jumping nonlinearities. Differential Integral Equations 14 (2001), no. 9, 1129--1152. https://projecteuclid.org/euclid.die/1356124311