Higher-order criteria and sharpness for blow-up at quantized singularities in Liouville equations
Abstract
This work is devoted to the study of blow-up phenomena for solutions of −Δu=λV(x)|x|2Nev, in B1, v=0 on ∂ B1. where B1 denotes the unit ball in ℝ2 centered at the origin, V(x) is a positive smooth function, N∈ ℕ and λ>0 is a small parameter. The main goal is to determine conditions on the potential V guaranteeing the existence of families of solutions that blow up at the origin as λ→ 0+. Moreover, we establish higher order criteria that distinguish between simple and non-simple blow-up behavior at x=0. In particular, for N−1∈ 2ℕ, we construct a non-simple blow-up sequence vk, corresponding to coefficient function Vk whose derivatives vanish up to the maximal order allowed by the upper bound of Wei-Zhang, thereby proving that this bound is sharp.
Related
- Integral Volume Criteria for Semilinear and Quasilinear Liouville Theorems on Complete Manifolds
- Asymptotic analysis and uniqueness of blowup solutions of non-quantized singular mean field equations
- Liouville Theorems for Stable Solutions to Anisotropic Quasilinear Equations
- Liouville Theorems for the Stationary Fractional Hall-Magnetohydrodynamic Equations
- Liouville Theorems for Semilinear Parabolic Equations Involving the Nonlinear Gradient Term
BibTeX
@misc{daprile2026higherorder,
title = {{Higher-order criteria and sharpness for blow-up at quantized singularities in Liouville equations}},
author = {Teresa D'Aprile and Juncheng Wei and Lei Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10968},
eprint = {2610.10968},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10968},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Higher-order criteria and sharpness for blow-up at quantized singularities in Liouville equations
AU - Teresa D'Aprile
AU - Juncheng Wei
AU - Lei Zhang
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10968
UR - https://arxiv.org/abs/2610.10968
AB - This work is devoted to the study of blow-up phenomena for solutions of −Δu=λV(x)|x|^2Ne^v, in B_1, v=0 on ∂ B_1. where B_1 denotes the unit ball in ℝ^2 centered at the origin, V(x) is a positive smooth function, N∈ ℕ and λ>0 is a small parameter. The main goal is to determine conditions on the potential V guaranteeing the existence of families of solutions that blow up at the origin as λ→ 0^+. Moreover, we establish higher order criteria that distinguish between simple and non-simple blow-up behavior at x=0. In particular, for N−1∈ 2^ℕ, we construct a non-simple blow-up sequence v_k, corresponding to coefficient function V_k whose derivatives vanish up to the maximal order allowed by the upper bound of Wei-Zhang, thereby proving that this bound is sharp.
ER -