2001 Half-eigenvalues of self-adjoint, $2m$th-order differential operators and semilinear problems with jumping nonlinearities
Bryan P. Rynne
Differential Integral Equations 14(9): 1129-1152 (2001). DOI: 10.57262/die/1356124311

## Abstract

We consider semilinear boundary value problems of the form $$L u(x) = f(x,u(x)) + h(x), \quad x \in (0,\pi), \tag*{(1)}$$ 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 $$L u = a u^+ - b u^- +{\lambda} u + h, \tag*{(2)}$$ 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.

## Citation

Bryan P. Rynne. "Half-eigenvalues of self-adjoint, $2m$th-order differential operators and semilinear problems with jumping nonlinearities." Differential Integral Equations 14 (9) 1129 - 1152, 2001. https://doi.org/10.57262/die/1356124311

## Information

Published: 2001
First available in Project Euclid: 21 December 2012

zbMATH: 1028.34075
MathSciNet: MR1852875
Digital Object Identifier: 10.57262/die/1356124311

Subjects:
Primary: 34B15
Secondary: 47J10

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