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.