A class of degenerate elliptic problems involving the Grushin operator: critical mass and ground states
Abstract
We study the following degenerate elliptic equation with combined nonlinearities and prescribed mass −Δγu+ωu+|u|p−2u=|u|q−2u in ℝN, ∫ℝN |u|2 dz=c, where Δγ is the Grushin operator and ω∈ℝ appears as an unknown Lagrange multiplier. 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.
Related
- Infinitely many solutions to critical problems involving the p-Grushin operator
- Elliptic problems involving a diffusion operator of vectorial p&q-Laplacian type
- Existence of solutions for a nonlinear elliptic equation involving (p(x),q(x))-Laplacian and (p(x),q(x))-biharmonic Operator
- Existence Results for the Critical Biharmonic Elliptic Equation Involving Singular Nonlinearities
- Anisotropic quasilinear elliptic systems involving nonlinearities with negative critical terms
BibTeX
@misc{jevnikar2026class,
title = {{A class of degenerate elliptic problems involving the Grushin operator: critical mass and ground states}},
author = {Aleks Jevnikar and Senli Liu and Yu Su},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36927},
eprint = {2610.36927},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36927},
}
RIS
TY - UNPB
TI - A class of degenerate elliptic problems involving the Grushin operator: critical mass and ground states
AU - Aleks Jevnikar
AU - Senli Liu
AU - Yu Su
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36927
UR - https://arxiv.org/abs/2610.36927
AB - We study the following degenerate elliptic equation with combined nonlinearities and prescribed mass −Δ_γu+ωu+|u|^p−2u=|u|^q−2u in ℝ^N, ∫_ℝ^N |u|^2 dz=c, where Δ_γ is the Grushin operator and ω∈ℝ appears as an unknown Lagrange multiplier. 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.
ER -