Nonlinear elliptic equations and systems with linear part at resonance

TL;DR

This paper extends classical results on resonance in nonlinear elliptic equations, providing new conditions for solution existence in systems based on spectral properties, and illustrating these with examples related to harmonic oscillators.

Contribution

It generalizes Williams' extension for systems at resonance, offering new spectral conditions and concrete examples linking classical oscillator results to elliptic systems.

Findings

Derived spectral conditions for solution existence in 2x2 systems

Connected classical oscillator results to elliptic system resonance

Provided examples illustrating the theoretical results

Abstract

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 $2 \times 2$ systems. We derive conditions for existence of solutions, which turned out to depend on the spectral properties of the linear part.