On Hamiltonian systems with critical Sobolev exponents

In this paper we consider lower order perturbations of the critical Lane-Emden system posed on a bounded smooth domain Ω⊂RN, with N≥3, inspired by the classical results of Brezis and Nirenberg [4]. We solve the problem of finding a positive solution for all dimensions N≥4. For the critical dimension...

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Bibliographic Details
Published in:Journal of Differential Equations 2023-07, Vol.360, p.314-346
Main Authors: Guimarães, Angelo, Moreira dos Santos, Ederson
Format: Article
Language:English
Subjects:
Critical dimension
Critical hyperbola
Lane-Emden systems
Positive solutions
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Summary:In this paper we consider lower order perturbations of the critical Lane-Emden system posed on a bounded smooth domain Ω⊂RN, with N≥3, inspired by the classical results of Brezis and Nirenberg [4]. We solve the problem of finding a positive solution for all dimensions N≥4. For the critical dimension N=3 we show a new phenomenon, not observed for scalar problems. Namely, there are parts on the critical hyperbola where solutions exist for all 1-homogeneous or subcritical superlinear perturbations and parts where there are no solutions for some of those perturbations.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2023.02.050