Potential Landesman-Lazer type conditions and the Fucik spectrum

Abstract

We prove the existence of solutions to the nonlinear problem u''(x) + λ+u+(x) = λ_u-(x) + g(x, u(x)) = ƒ(x), x ∈ (0, π), u(0) = u(π) = 0 where the point [λ+, λ_] is a point of the Fučík spectrum and the nonlinearity g(x, u(x)) satisfies a potential Landesman-Lazer type condition. We use a variational method based on the organization of the Saddle Point Theorem.