Differential and Integral Equations

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

Bryan P. Rynne

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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

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

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34B15: Nonlinear boundary value problems
Secondary: 47J10: Nonlinear spectral theory, nonlinear eigenvalue problems [See also 49R05]


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

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