Electron. J. Diff. Equ., Vol. 2011 (2011), No. 112, pp. 1-34.

Periodic boundary-value problems and the Dancer-Fucik spectrum under conditions of resonance

David A. Bliss, James Buerger, Adolfo J. Rumbos

Abstract:
We prove the existence of solutions to the nonlinear $2 \pi$-periodic problem
$$\displaylines{ u''(x)+\mu u^+(x)-\nu u^-(x)+g(x,u(x))=f(x)\,,\quad x\in (0,2\pi)\,,\cr u(0)-u(2\pi) =0 \,, \quad u'(0) - u'(2\pi)=0, }$$
where the point $(\mu,\nu)$ lies in the Dancer-Fucik spectrum, with
$$ 0< \frac{4}{9}\mu \leqslant \nu<\mu \quad\hbox{and}\quad \mu<(m+1)^2, $$
for some natural number m, and the nonlinearity $g(x,\xi)$ is bounded with primitive, $G(x,\xi)$, satisfying a Landesman-Lazer type condition introduced by Tomiczek in 2005. We use variational methods based on the generalization of the Saddle Point Theorem of Rabinowitz.

Submitted November 10, 2010. Published August 29, 2011.
Math Subject Classifications: 34B15, 34K13, 35A15.
Key Words: Resonance; jumping nonlinearities; Dancer-Fucik spectrum; saddle point theorem.

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David A. Bliss
School of Mathematical Sciences, Claremont Graduate University
Claremont, CA 91711, USA
david.a.bliss@jpl.nasa.gov
  James Buerger
Department of Mathematics, Pomona College
Claremont, CA 91711, USA
email: jbuerg127@gmail.com
Adolfo J. Rumbos
Department of Mathematics, Pomona College
Claremont, CA 91711, USA
email: arumbos@pomona.edu

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