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\title{\vskip-2.5cm
{Rich dynamics in planar systems with heterogeneous nonnegative weights}
\thanks{This paper has been written under the auspices of the Ministry of
Science, Technology and Universities of Spain, under Research
Grants PGC2018-097104-B-I00 and PID2021-123343NB-I00, and of the IMI of Complutense
University. } }


\date{\today}




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\begin{document}

\maketitle

{

\begin{abstract}
This paper studies the global structure of the set of nodal solutions of a generalized
Sturm--Liouville boundary value problem associated to the quasilinear  equation
$$
-(\phi(u'))'= \lambda u + a(t)g(u), \quad \lambda\in {\mathbb R},
$$
where $a(t)$ is non-negative with some positive humps separated away by
intervals of degeneracy where $a\equiv 0$. When $\phi(s)=s$ this equation includes
a generalized prototype of a classical model going back to Moore and Nehari \cite{MN-1959}, 1959.
This is the first paper where the general case when $\l\in\R$ has been addressed when $a\gneq 0$.
The semilinear case with $a\lneq 0$ has been recently treated by L\'{o}pez-G\'{o}mez and Rabinowitz
\cite{LGR-2015,LGR-2017,LGR-2020}.
\\
\\
{\it 2010 Mathematics Subject Classification:} 34B08, 34B24, 35C15.
\\
\\
{\it Keywords and Phrases}. Nonlinear differential equations, planar systems,
degenerate weights, Moore--Nehari equation, nodal solutions, global bifurcation theory,
a priori bounds, Poincar\'{e} maps.
\end{abstract}

}





This
\end{document}