Electronic Journal of Differential Equations,
Vol. 2000(2000), No. 37, pp. 1-21.
Title: Resonance with respect to the Fucik spectrum
Authors: A. K. Ben-Naoum (Univ. de Mons - Hainaut, Belgium)
C. Fabry (Univ. catholique de Louvain, Belgium)
D. Smets (Univ. catholique de Louvain, Belgium)
Abstract:
Let $L$ be a self-adjoint operator on $L^2(\Omega; R)$ with
$\Omega$ a bounded and
open subset of $R^N$. This article considers the resonance problem
with respect to the Fu\v c\'\i k spectrum of $L$, which means that
we study equations of the form
$$
Lu = \alpha u^+ - \beta u^- + f(\cdot,u),
$$
when the homogeneous equation $Lu = \alpha u^+ - \beta u^-$
has non-trivial solutions. Using the computation of degrees that are
not necessarily $+1$ or $-1$, we present results about the
existence of solutions. Our results are illustrated
with examples and can be seen as generalizations of Landesman-Lazer
conditions. Non-existence results are also given.
Submitted January 4, 2000. Published May 17, 2000.
Math Subject Classifications: 70K30, 35P30, 35G30.
Key Words: Resonance; jumping nonlinearity; Landesman-Lazer conditions.