Electron. J. Diff. Eqns., Vol. 2000(2000), No. 37, pp. 1-21.

Resonance with respect to the Fucik spectrum

A. K. Ben-Naoum, C. Fabry, & D. Smets

Abstract:
Let $L$ be a self-adjoint operator on $L^2(\Omega;\Bbb R)$ with $\Omega$ a bounded and open subset of ${\Bbb R}^N$. This article considers the resonance problem with respect to the Fucik 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.

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A. K. Ben-Naoum
Universite de Mons - Hainaut
Institut de mathematique et d'informatique
6 Avenue du Champ de Mars, B - 7000 Mons, Belgium
e-mail: kbn@bsb.be
C. Fabry
Universite catholique de Louvain
Departement de mathematique
2 Chemin du cyclotron, B - 1348 Louvain-la-Neuve, Belgium
e-mail: fabry@math.ucl.ac.be
Didier Smets
Universite catholique de Louvain
Departement de mathematique
2 Chemin du cyclotron, B - 1348 Louvain-la-Neuve, Belgium
e-mail: smets@amm.ucl.ac.be

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