Heriot-Watt Mathematics Report Series
HWM00-1, 25 Feb 2000
Half-eigenvalues of self-adjoint, 2mth order differential operators and semilinear problems with jumping nonlinearities
B P Rynne
Abstract
We consider semilinear boundary value problems
of the form
\begin{equation} \label{abs_orig.eq}
L u(x) = f(x,u(x)) + h(x), \quad x \in (0,\pi),
\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
$f : [0,\pi] \X \R \tends \R$ is a Carath{é}odory function
and $h \in L^2(0,\pi)$.
We assume that the limits
$$
a(x) := \lim_{\xi \tends \infty} f(x,\xi)/\xi,
\quad
b(x) := \lim_{\xi \tends -\infty} f(x,\xi)/\xi,
$$
exist for a.e.~$x \in [0,\pi]$ and $a,\,b \in L^{\infty}(0,\pi)$,
but $a \ne b$.
In this case the nonlinearity $f$ is termed jumping}.
Closely related to \eqref{abs_orig.eq} is the `limiting' boundary
value problem
\begin{equation} \label{abs_heval.eq}
L u = a u^+ - b u^- +\la u + h, % \qquad \mbox{in } (0,\pi),
\end{equation}
where
$u^{\pm}(x) = \max\{\pm u(x),0\}$ for $x \in [0,\pi]$,
and $\la$ is a real parameter.
Values of $\la$ for which \eqref{abs_heval.eq} (with $h=0$) has a
non-trivial solution $u$ will be called 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 \eqref{abs_orig.eq} and \eqref{abs_heval.eq}.
These result depend strongly on the location of the half-eigenvalues
relative to the point $\la=0$.
Some geometric properties of the
Fucik spectrum of $L$ are also briefly discussed.
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