Resonance with Respect to the Fucik Spectrum
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Let L be a self-adjoint operator on L2(Ω;ℝ) with Ω a bounded and open subset of ℝN. This article considers the resonance problem with respect to the Fučík spectrum of L, which means that we study equations of the form Lu = αu+ - βu- + ƒ(•,u), when the homogeneous equation Lu = αu+ - βu- 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 Landseman-Lazer conditions. Non-existence results are also given.