Electron. J. Diff. Equ., Vol. 2016 (2016), No. 67, pp. 1-17.

Nonlinear elliptic equations and systems with linear part at resonance

Philip Korman

The famous result of Landesman and Lazer [10] dealt with resonance at a simple eigenvalue. Soon after publication of [10], Williams [14] gave an extension for repeated eigenvalues. The conditions in Williams [14] are rather restrictive, and no examples were ever given. We show that seemingly different classical result by Lazer and Leach [11], on forced harmonic oscillators at resonance, provides an example for this theorem. The article by Williams [14] also contained a shorter proof. We use a similar approach to study resonance for 2X2 systems. We derive conditions for existence of solutions, which turned out to depend on the spectral properties of the linear part.

Submitted March 27, 2015. Published March 10, 2016.
Math Subject Classifications: 35J61, 35J47.
Key Words: Elliptic systems at resonance; resonance at multiple eigenvalues; Lazer and Leach condition.

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Philip Korman
Department of Mathematical Sciences
University of Cincinnati
Cincinnati Ohio 45221-0025, USA
email: kormanp@ucmail.uc.edu

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