Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 112, pp. 1-34.
Title: Periodic boundary-value problems and the Dancer-Fucik
spectrum under conditions of resonance
Authors: David A. Bliss (Claremont Graduate Univ., Claremont, CA, USA)
James Buerger (Pomona College, Claremont, CA, USA)
Adolfo J. Rumbos (Pomona College, Claremont, CA, USA)
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.