Nonlinear elliptic equations and systems with linear part at resonance

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.