Resonance with respect to the Fucik spectrum

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 Fuv 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.

Keywords

• Resonance
• jumping nonlinearity
• Landesman-Lazer conditions.