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\documentclass[12pt]{article}
\begin{document}
\title{Sign-changing solutions of singular Dirichlet boundary
value problems }
\author{Irena Rach\accent23 unkov\'a  and Svatoslav Stan\v ek}
\maketitle
\noindent
We consider the singular
Dirichlet boundary value problem
\begin{equation}                                    \label{r1.1}
(r(x(t))x'(t))'=\mu q(t) f(t,x(t)),
\end{equation}
\begin{equation}                                    \label{r1.2}
x(0)=x(T)=0, \ \max\{x(t): 0 \le t\le T \}\cdot \min\{x(t): 0 \le t
\le T\}<0,
\end{equation}
where $\mu$ is a positive parameter and $f$ is singular at the
point $x=0$ of the phase variable $x$ in the following sense
\begin{equation}                                     \label{r1.3}
\lim_{x \to 0^-}f(t,x)=-\infty, \ \lim_{x \to 0^+}f(t,x)=\infty
\quad \mbox{for} \ t \in [0,T].
\end{equation}

We say that a function $x \in C^1([0, T])$ is a {\it solution of
problem} (\ref{r1.1}),\,(\ref{r1.2}) if $x$ has precisely one
zero $t_0$ in $(0,T)$, $r(x)x' \in C^1((0,T) \setminus \{t_0\})$,
$x$ fulfils (\ref{r1.2}) and there exists $\mu_0>0$ such that
(\ref{r1.1}) is satisfied for $\mu=\mu_0$ and $t \in
(0,T)\setminus \{t_0\}$.

We have found effective conditions for
the functions $r,\,q$ and $f$ which guarantee the existence of
solutions to problem (\ref{r1.1}),\,(\ref{r1.2}). Any such
solution goes through the singularity of $f$. As far as we know,
this case has not been solved yet. Up till now, only positive
(negative) solutions on $(0,T)$ of the Dirichlet problem with
the singularity at the point $x=0$ of the phase variable $x$ in
nonlinearities of considered second-order differential equations
have been studied.

\begin{thebibliography}{99}
\bibitem{rs1} 
I. Rach\accent23 unkov\'a and S. Stan\v ek, Sign-changing solutions of singular Dirichlet boundary value problems, 
\textit{J. Inequal. Appl.}, to appear.

\end{thebibliography}
\end{document}