Electron. J. Diff. Eqns., Vol. 2005(2005), No. 94, pp. 1-12.

Potential Landesman-Lazer type conditions and the Fucik spectrum

Petr Tomiczek

Abstract:
We prove the existence of solutions to the nonlinear problem
$$\displaylines{ u''(x)+\lambda_+ u^+(x)-\lambda_- u^-(x)+g(x,u(x))=f(x)\,,\quad x\in (0,\pi)\,,\cr u(0)=u(\pi)=0 } $$
where the point $[\lambda_+,\lambda_-]$ is a point of the Fucik spectrum and the nonlinearity $g(x,u(x))$ satisfies a potential Landesman-Lazer type condition. We use a variational method based on the generalization of the Saddle Point Theorem.

Submitted November 26, 2004. Published August 29, 2005.
Math Subject Classifications: 35J70, 58E05, 49B27.
Key Words: Resonance; eigenvalue; jumping nonlinearities; Fucik spectrum.

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Petr Tomiczek
Department of Mathematics
University of West Bohemia
Univerzitni 22, 306 14 Plzen, Czech Republic
e-mail: tomiczek@kma.zcu.cz

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