Electronic Journal of Differential Equations,
Conf. 10 (2002), pp. 101--107.
Title: A sign-changing solution for a superlinear Dirichlet problem, II
Authors: Alfonso Castro (Univ. of Texas at San Antonio, USA)
Pavel Drabek (Univ. of West Bohemia, Pilsen, Czech Republic)
John M. Neuberger (Northern Arizona Univ.,Flagstaff, AZ, USA)
Abstract:
In previous work by Castro, Cossio, and Neuberger \cite{ccn},
it was shown that a superlinear Dirichlet problem has
at least three nontrivial solutions when the derivative of the
nonlinearity at zero is less than the first eigenvalue of
$-\Delta$ with zero Dirichlet boundry condition.
One of these solutions changes sign exactly-once and the other
two are of one sign.
In this paper we show that when this derivative is
between the $k$-th and $k+1$-st eigenvalues there still
exists a solution which changes sign at most $k$ times.
In particular, when $k=1$ the sign-changing {\it exactly-once}
solution persists although one-sign solutions no longer exist.
Published February 28, 2003.
Math Subject Classifications: 35J20, 35J25, 35J60.
Key Words: Dirichlet problem; superlinear; subcritical;
sign-changing solution; deformation lemma.