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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.

MSC:
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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