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Nonlinear elliptic equations and systems with linear part at resonance. (English) Zbl 1383.35095
Summary: The famous result of E. M. Landesman and A. C. Lazer [J. Math. Mech. 19, 609–623 (1970; Zbl 0193.39203)] dealt with resonance at a simple eigenvalue. Soon after publication of [loc. cit.], S. A. Williams [J. Differ. Equations 8, 580–586 (1970; Zbl 0209.13003)] gave an extension for repeated eigenvalues. The conditions in [Williams, loc. cit.] are rather restrictive, and no examples were ever given. We show that seemingly different classical result by A. C. Lazer and D. E. Leach [Ann. Mat. Pura Appl. (4) 82, 49–68 (1969; Zbl 0194.12003)], on forced harmonic oscillators at resonance, provides an example for this theorem. The article by Williams [loc. cit.] 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.

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J57 Boundary value problems for second-order elliptic systems
35J61 Semilinear elliptic equations
35J47 Second-order elliptic systems
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