preprint

Inserted: 28 oct 2026
Last Updated: 28 oct 2026

Year: 2026

Abstract:

We study a degenerate elliptic equation with combined nonlinearities and prescribed mass involving the Grushin operato. We characterize the critical mass threshold and the energy ground state solutions. Firstly, we prove the existence of optimizers for the Gagliardo-Nirenberg inequality involving the Grushin operator together with estimates for the related best constant. Secondly, we characterize the critical mass by the best constant of Gagliardo-Nirenberg inequality and prove the existence of $x$-radial energy ground state solutions. Finally, we describe the relation between the $x$-radial action and energy ground state solutions, and the optimizers of Gagliardo-Nirenberg inequality. The strategy developed here is based on the interplay between the mass-preserving scaling and a sharp Gagliardo-Nirenberg inequality. It therefore suggests possible extensions to other prescribed-mass problems with competing homogeneous terms and threshold-mass phenomena.