Optimal multipolar Lp-Hardy inequalities: ground states, criticality and variational thresholds
Abstract
We study ground states, criticality, and variational compactness for genuinely multipolar Lp-Hardy weights, 1<p<n. A ground-state representation across the poles gives a complete criticality classification of the full potential and characterizes finite-energy attainment. Removing a geometric interaction term produces a variational problem governed by concentration at the poles and infinity. A strict concentration gap yields compactness, positive extremizers, and criticality of the optimal operator; for p≥2, it is also necessary for attainment. In the subquadratic range, the optimal truncated coefficient is below one and attained throughout the full-potential critical set. For admissible parameters with 1<p<2 and (n−p)/p<s≤ m(n−p)/p, arbitrary fixed pole configurations exhibit negative directions, attained critical states, and coercive subcritical regimes outside this set along positive-weight paths towards the simplex vertices. A single-pole degeneration formula and continuity in the weights establish this coexistence. In the superquadratic range, an interaction threshold determines a non-attained infinity-concentration interval. Under strict separation from pole concentration, its endpoint is critical with infinite energy and a compact phase follows. We prove separation for every bipolar configuration and in several multipolar regimes, including a vertex neighborhood uniform in the exponent. Near parameters strictly inside the quadratic attained region, the optimal coefficient crosses one continuously through p=2. The proofs combine ground-state remainders, capacity, weighted compactness, and critical Hardy barriers.
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BibTeX
@misc{jin2026optimal,
title = {{Optimal multipolar Lp-Hardy inequalities: ground states, criticality and variational thresholds}},
author = {Yongyang Jin and Shoufeng Shen and Li Tang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00652},
eprint = {2610.00652},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00652},
}
RIS
TY - UNPB
TI - Optimal multipolar Lp-Hardy inequalities: ground states, criticality and variational thresholds
AU - Yongyang Jin
AU - Shoufeng Shen
AU - Li Tang
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.00652
UR - https://arxiv.org/abs/2610.00652
AB - We study ground states, criticality, and variational compactness for genuinely multipolar L^p-Hardy weights, 1<p<n. A ground-state representation across the poles gives a complete criticality classification of the full potential and characterizes finite-energy attainment. Removing a geometric interaction term produces a variational problem governed by concentration at the poles and infinity. A strict concentration gap yields compactness, positive extremizers, and criticality of the optimal operator; for p≥2, it is also necessary for attainment. In the subquadratic range, the optimal truncated coefficient is below one and attained throughout the full-potential critical set. For admissible parameters with 1<p<2 and (n−p)/p<s≤ m(n−p)/p, arbitrary fixed pole configurations exhibit negative directions, attained critical states, and coercive subcritical regimes outside this set along positive-weight paths towards the simplex vertices. A single-pole degeneration formula and continuity in the weights establish this coexistence. In the superquadratic range, an interaction threshold determines a non-attained infinity-concentration interval. Under strict separation from pole concentration, its endpoint is critical with infinite energy and a compact phase follows. We prove separation for every bipolar configuration and in several multipolar regimes, including a vertex neighborhood uniform in the exponent. Near parameters strictly inside the quadratic attained region, the optimal coefficient crosses one continuously through p=2. The proofs combine ground-state remainders, capacity, weighted compactness, and critical Hardy barriers.
ER -