Asymptotic estimates and nonexistence results for critical problems with Hardy term involving Grushin-type operators

  • Annunziata LoiudiceEmail author


We provide the asymptotic behavior of solutions, at the singularity and at infinity, for a class of subelliptic Dirichlet problems with Hardy perturbation and critical nonlinearity of the type \(-\,{\mathcal {L}}_\alpha u -\mu \dfrac{\psi ^2}{d^2} u =K(z)|u|^{2^*-2}u\) in \(\varOmega \), where \({\mathcal {L}}_\alpha = \varDelta _x + |x|^{2\alpha }\varDelta _y\), \(\alpha >0\) is the so-called Grushin operator, \(\varOmega \) is an open subset of \({\mathbb {R}}^N\), \(0\in \varOmega \), d is the gauge norm naturally associated with \({\mathcal {L}}_\alpha \), \(\psi :=|\nabla _\alpha d|\), where \(\nabla _\alpha \) is the Grushin gradient, \(K\in L^\infty \) and \(0\le \mu < {\overline{\mu }}\), where \({\overline{\mu }}\) is the best Hardy constant for \({\mathcal {L}}_\alpha \). Furthermore, we establish some Pohozaev-type nonexistence results.


Subelliptic critical problem Grushin operator Hardy potential Asymptotic behavior Pohozaev-type identity Nonexistence results 

Mathematics Subject Classification

35J70 35J75 35B40 



The author is a member of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), and she is partially supported by the GNAMPA research Project 2018 “Metodi di analisi armonica e teoria spettrale per le equazioni dispersive.”


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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di BariBariItaly

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