Heriot-Watt Mathematics Report Series
HWM02-39, 26 Dec 2002
Second order, Sturm-Liouville problems with asymmetric, superlinear nonlinearities II
B P Rynne
Abstract
We consider the nonlinear Sturm-Liouville problem
\begin{gather*}
-(p(x) u'(x))' +q(x) u(x) = f(x,u(x)) + h(x) , \quad \mbox{in } (0,\pi),
\label{abs_main.eq}\\
c_{00} u(0) + c_{01} u'(0) = 0,
\qquad
c_{10} u(\pi) + c_{11} u'(\pi) = 0,
\label{abs_O1.eq}
\end{gather*}
where:
$p \in C^1[0,\pi]$, $q \in C^0[0,\pi]$,
with $p(x) > 0$ for all $x \in [0,\pi]$;
$c_{i0}^2+c_{i1}^2 > 0$, $i=0,\,1$;
$h \in L^2(0,\pi)$.
We suppose that $f : [0,\pi] \times R \tends R$ is continuous and
there exist increasing functions
$\zeta_l,\,\zeta_u : [0,\infty) \tends R$,
and positive constants $A$, $B$, such that
$\lim_{t \tends \infty} \zeta_l(t) = \infty$
and
\begin{alignat*}{2}
-A + \zeta_l(\xi) \xi \le f(x,\xi) &\le A + \zeta_u(\xi) \xi ,
&\quad&\xi \ge 0 ,
\\
|f(x,\xi)| &\le A + B |\xi| ,
&&\xi \le 0 ,
\end{alignat*}
for all $(x,\eta) \in [0,\pi] \times R$
(thus the nonlinearity is
superlinear as $u(x) \tends \infty$,
but linearly bounded as $u(x) \tends -\infty$).
Existence and non-existence results are obtained for the above
problem.Similar results have been obtained before for problems in which
$f$ is linearly bounded as $|\xi| \to \infty$, and these results have
been expressed in terms of `half-eigenvalues' of the problem.
The results obtained here for the superlinear case are expressed in ms
of certain asymptotes of these half-eigenvalues.
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