Phase transition, sharp asymptotics and symmetry for critical quasilinear Hartree equations beyond σ=2p
Abstract
We establish the sharp asymptotic and symmetry theory for the critical quasilinear Hartree equation −Δpu=(|x|−σ*up*σ)up*σ−1 in ℝN, p*σ=(p(2N−σ))/(2(N−p)), in the complementary range 2p<σ<N, beyond the quasilinear theory previously available for σ≤2p [Math. Ann., 391(2):2653--2708, 2025]. This range is structurally different: p*σ<p, so the effective potential obtained by factoring out up−1 becomes singular as u→0. Nevertheless, whenever p*σ>1, every nonnegative finite-energy weak solution is automatically locally bounded. Our proof provides a quasilinear regularity mechanism adapted to this complementary range, where the potential-theoretic route available for σ≤2p breaks down. For positive finite-energy solutions with p*σ>1 we moreover uncover a sharp transition governed by the sign of σ(p−2)+2p. If this quantity is positive, the classical p-harmonic profile persists. At the threshold, u(x)≃ |x|−γh(log|x|)1/(p−p*σ), γh=(N−p)/(p−1), with the matching two-sided gradient rate. Below the threshold, u(x)=A*|x|−γ*+o(|x|−γ*), γ*=(2(N−p)(σ−p))/(p(σ−2p))<γh, and the rescaled gradients converge uniformly to the corresponding radial profile. In all three regimes the solutions are radially symmetric about some point and strictly decreasing in the radial variable.
Related
- Systems of Kirchhoff quasilinear Schrodinger equations with Hardy potential and critical Choquard-type nonlinearity
- Doubly critical mixed-order quasilinear equations: existence, multiplicity and regularity
- Global c1,α regularity and fine behavior at critical points for quasilinear elliptic equations with Orlicz growth and unbounded drift
- Symmetry, monotonicity, and asymptotics of singular solutions to semilinear elliptic equations
- Steady-State Phase Transition States and Large-Time Asymptotics for the Cauchy Problem of Compressible Navier-Stokes Equations with van der Waals Equation of State
BibTeX
@misc{le2026phase,
title = {{Phase transition, sharp asymptotics and symmetry for critical quasilinear Hartree equations beyond $σ=2p$}},
author = {Phuong Le},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00339},
eprint = {2610.00339},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00339},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Phase transition, sharp asymptotics and symmetry for critical quasilinear Hartree equations beyond σ=2p
AU - Phuong Le
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00339
UR - https://arxiv.org/abs/2610.00339
AB - We establish the sharp asymptotic and symmetry theory for the critical quasilinear Hartree equation −Δ_pu=(|x|^−σ*u^p^*_σ)u^p^*_σ−1 in ℝ^N, p^*_σ=(p(2N−σ))/(2(N−p)), in the complementary range 2p<σ<N, beyond the quasilinear theory previously available for σ≤2p [Math. Ann., 391(2):2653--2708, 2025]. This range is structurally different: p^*_σ<p, so the effective potential obtained by factoring out u^p−1 becomes singular as u→0. Nevertheless, whenever p^*_σ>1, every nonnegative finite-energy weak solution is automatically locally bounded. Our proof provides a quasilinear regularity mechanism adapted to this complementary range, where the potential-theoretic route available for σ≤2p breaks down. For positive finite-energy solutions with p^*_σ>1 we moreover uncover a sharp transition governed by the sign of σ(p−2)+2p. If this quantity is positive, the classical p-harmonic profile persists. At the threshold, u(x)≃ |x|^−γ_h(log|x|)^1/(p−p^*_σ), γ_h=(N−p)/(p−1), with the matching two-sided gradient rate. Below the threshold, u(x)=A_*|x|^−γ_*+o(|x|^−γ_*), γ_*=(2(N−p)(σ−p))/(p(σ−2p))<γ_h, and the rescaled gradients converge uniformly to the corresponding radial profile. In all three regimes the solutions are radially symmetric about some point and strictly decreasing in the radial variable.
ER -