On the classification of solutions to an elliptic equation involving the Grushin operator

Abstract

We prove that, for , the degenerate elliptic equation involving the Grushin operator on the whole space

does not admit any solution stable outside a compact set of . The result is sharp and obtained without any assumption about the boundedness of solutions. In particular, when , we recover the previous results for the Laplace operator in [Farina, C R Math Acad Sci Paris. 2007;345:63–66] and [Dancer and Farina, Proc Am Math Soc.2009;137:1333–1338].

Keywords:

• Grushin operator
• solutions stable outside a compact set
• test functions method
• Grushin spherical function
• integral estimates
• contradiction arguments

AMS Subject Classifications:

• 35J70
• 35B05

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported by Vietnam Ministry of Education and Training [grant number B2016-SPH-17].