A priori bounds for energy-bounded solutions of critical polyharmonic equations
Abstract
We investigate critical polyharmonic equations of the type: Lu = |u|2♯−2 u in Ω with Dirichlet boundary conditions, in a smooth bounded domain Ω of ℝn. Here L is an elliptic differential operator of integer order 2 ≤ 2k < n whose leading order term is (−Δ)k and 2♯ = 2n/(n−2k) is the critical Sobolev exponent. Our main result establishes, in large dimensions, uniform \emph{a priori} bounds in C2k(Ω̅) for bounded-energy solutions of this problem, that only depend on an upper bound on the energy. We prove this under a coercivity assumption of sorts on the lower-order terms of L. Our results are sharp, at least when k=1. Our approach uses asymptotic analysis techniques and in the course of the proof we obtain in particular a new global pointwise description of bounded-energy blowing-up solutions for this problem, which is of independent interest.
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BibTeX
@misc{carletti2026priori,
title = {{A priori bounds for energy-bounded solutions of critical polyharmonic equations}},
author = {Lorenzo Carletti and Bruno Premoselli},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.29690},
eprint = {2605.29690},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.29690},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A priori bounds for energy-bounded solutions of critical polyharmonic equations
AU - Lorenzo Carletti
AU - Bruno Premoselli
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2605.29690
UR - https://arxiv.org/abs/2605.29690
AB - We investigate critical polyharmonic equations of the type: Lu = |u|^2^♯−2 u in Ω with Dirichlet boundary conditions, in a smooth bounded domain Ω of ℝ^n. Here L is an elliptic differential operator of integer order 2 ≤ 2k < n whose leading order term is (−Δ)^k and 2^♯ = 2n/(n−2k) is the critical Sobolev exponent. Our main result establishes, in large dimensions, uniform \emph{a priori} bounds in C^2k(Ω̅) for bounded-energy solutions of this problem, that only depend on an upper bound on the energy. We prove this under a coercivity assumption of sorts on the lower-order terms of L. Our results are sharp, at least when k=1. Our approach uses asymptotic analysis techniques and in the course of the proof we obtain in particular a new global pointwise description of bounded-energy blowing-up solutions for this problem, which is of independent interest.
ER -