Ninth MSU-UAB Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 20 (2013), pp. 103-117.

A Landesman-Lazer condition for the boundary-value problem $-u''=a u^+ - b u^- +g(u)$ with periodic boundary conditions

Quinn A. Morris, Stephen B. Robinson

Abstract:
In this article we prove the existence of solutions for the boundary-value problem
$$\displaylines{ -u''=a u^+ - b u^- +g(u)\cr u(0)=u(2 \pi)\cr u'(0)=u'(2 \pi), }$$
where $(a,b)\in \mathbb{R}^2$, $u^+ (x) = \max \{u(x),0\}$, $u^- (x) = \max \{-u(x),0\}$, and $g: \mathbb{R} \to \mathbb{R}$ is a bounded, continuous function. We consider both the resonance and nonresonance cases relative to the Fucik Spectrum. For the resonance case we assume a generalized Landesman-Lazer condition that depends upon the average values of g at $\pm\infty$. Our theorems generalize the results in [1] by removing certain restrictions on (a,b). Our proofs are also different in that they rely heavily on a variational characterization of the Fucik Spectrum given in [3].

Published October 31, 2013.
Math Subject Classifications: 34B15.
Key Words: Fucik spectrum; resonance; Landesman-Lazer condition; variational approach.

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Quinn A. Morris
Department of Mathematics and Statistics
The University of North Carolina at Greensboro
116 Petty Building, 317 College Avenue
Greensboro, NC 27412, USA
email: qamorris@uncg.edu
Stephen B. Robinson
Department of Mathematics, Wake Forest University
PO Box 7388, 127 Manchester Hall
Winston-Salem, NC 27109, USA
email: sbr@wfu.edu

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