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

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.