Balance between degenerate elliptic operators and coercive Hamiltonians
Abstract
For p>1, we consider the boundary value problem for fully nonlinear degenerate elliptic equations −λi(D2u)+|Du|p+γu=f(x) in bounded domains with Dirichlet or boundary blow-up conditions; here λi(D2u) denotes the i-th eigenvalue of the Hessian. We study existence and nonexistence of solutions together with the asymptotic behaviour of the solutions when γ goes to zero. A priori Lipschitz estimates play an important role. The interplay between the operator's degeneracy and the superlinear growth of the Hamiltonian gives rise to phenomena that are very different depending on which of the two terms dominates, e.g. the ergodic dichotomy takes place only when i=N, while new phenomena arise for i<N in which case, under mild conditions, solutions that blow up even in just one point do not exist, and conditions on the size of f must be imposed for the existence of solutions to the Dirichlet problem with homogeneous boundary condition.
Related
- Sharp Riesz and reverse Riesz inequalities for degenerate elliptic operators of Grushin type
- On a Nonuniformly Coercive Quasilinear p(·)-Elliptic Neumann Problem with L1 Data
- On ABP Estimates for a Class of Fully Nonlinear Singular-Degenerate Elliptic Equations and Applications
- Unique continuation, nonexistence and bubbling for the critical p-Laplace equation II: the degenerate elliptic case and higher dimensions
- A class of degenerate elliptic problems involving the Grushin operator: critical mass and ground states
BibTeX
@misc{birindelli2026balance,
title = {{Balance between degenerate elliptic operators and coercive Hamiltonians}},
author = {Isabeau Birindelli and Giulio Galise and Hitoshi Ishii},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.02198},
eprint = {2608.02198},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.02198},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Balance between degenerate elliptic operators and coercive Hamiltonians
AU - Isabeau Birindelli
AU - Giulio Galise
AU - Hitoshi Ishii
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2608.02198
UR - https://arxiv.org/abs/2608.02198
AB - For p>1, we consider the boundary value problem for fully nonlinear degenerate elliptic equations −λ_i(D^2u)+|Du|^p+γu=f(x) in bounded domains with Dirichlet or boundary blow-up conditions; here λ_i(D^2u) denotes the i-th eigenvalue of the Hessian. We study existence and nonexistence of solutions together with the asymptotic behaviour of the solutions when γ goes to zero. A priori Lipschitz estimates play an important role. The interplay between the operator's degeneracy and the superlinear growth of the Hamiltonian gives rise to phenomena that are very different depending on which of the two terms dominates, e.g. the ergodic dichotomy takes place only when i=N, while new phenomena arise for i<N in which case, under mild conditions, solutions that blow up even in just one point do not exist, and conditions on the size of f must be imposed for the existence of solutions to the Dirichlet problem with homogeneous boundary condition.
ER -