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



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 \(\mathbb {G}\).


Subelliptic critical problem Hardy potential asymptotic behavior Carnot groups 

Mathematics Subject Classification

35J70 35J75 35B40 


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Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di BariBariItaly

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