Local Behavior of Solutions to Subelliptic Problems with Hardy Potential on Carnot Groups

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Abstract

We determine the exact behavior at the singularity of solutions to semilinear subelliptic problems of the type \(-\Delta _{\mathbb {G}}u -\mu \dfrac{\psi ^2}{d^2} u =f(\xi ,u)\) in \(\Omega \), \(u=0\) on \(\partial \Omega \), where \(\Delta _{\mathbb {G}}\) is a sub-Laplacian on a Carnot group \(\mathbb {G}\) of homogeneous dimension Q, \(\Omega \) is an open subset of \(\mathbb {G}\), \(0\in \Omega \), d is the gauge norm on \(\mathbb {G}\), \(\psi :=|\nabla _{\mathbb {G}}d|\), where \(\nabla _{\mathbb {G}}\) is the horizontal gradient associated with \(\Delta _{\mathbb {G}}\), f has at most critical growth and \(0\le \mu < \overline{\mu }\), where \(\overline{\mu }=\left( \frac{Q-2}{2} \right) ^2\) is the best Hardy constant on \(\mathbb {G}\).

Keywords

Subelliptic critical problem Hardy potential asymptotic behavior Carnot groups 

Mathematics Subject Classification

35J70 35J75 35B40 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di BariBariItaly

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