Positive radial ground states for nonlinear biharmonic equations: maximum principles and expanding-domain approximation
Abstract
We prove the existence of nonnegative radial ground states for a class of nonlinear biharmonic equations in ℝN, covering subcritical and critical nonlinearities. The solutions are obtained as limits of clamped ground states on expanding balls. The Cassani-Tarsia homogeneous maximum principle in [D. Cassani. A. Tarsia, Adv. Nonlinear Anal. 11 (2022)] makes the approximating states positive. A variational decomposition on the radial Nehari manifold then excludes every zero sphere of positive radius, so the limiting ground state is positive on ℝN∖{0}. When the fourth order operator factorises into two second-order operators with positive resolvents, the conclusion upgrades to u>0 throughout ℝN, including at the origin. We also give an explicit counterexample showing that the tempting extension of the homogeneous maximum principle to super-solutions with arbitrary nonhomogeneous boundary data is false, even with an arbitrarily large positive zeroth-order coefficient.
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BibTeX
@misc{cassani2026positive,
title = {{Positive radial ground states for nonlinear biharmonic equations: maximum principles and expanding-domain approximation}},
author = {Daniele Cassani and Zhisu Liu and Giulio Romani and Antonio Tarsia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11773},
eprint = {2609.11773},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11773},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Positive radial ground states for nonlinear biharmonic equations: maximum principles and expanding-domain approximation
AU - Daniele Cassani
AU - Zhisu Liu
AU - Giulio Romani
AU - Antonio Tarsia
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.11773
UR - https://arxiv.org/abs/2609.11773
AB - We prove the existence of nonnegative radial ground states for a class of nonlinear biharmonic equations in ℝ^N, covering subcritical and critical nonlinearities. The solutions are obtained as limits of clamped ground states on expanding balls. The Cassani-Tarsia homogeneous maximum principle in [D. Cassani. A. Tarsia, Adv. Nonlinear Anal. 11 (2022)] makes the approximating states positive. A variational decomposition on the radial Nehari manifold then excludes every zero sphere of positive radius, so the limiting ground state is positive on ℝ^N∖{0}. When the fourth order operator factorises into two second-order operators with positive resolvents, the conclusion upgrades to u>0 throughout ℝ^N, including at the origin. We also give an explicit counterexample showing that the tempting extension of the homogeneous maximum principle to super-solutions with arbitrary nonhomogeneous boundary data is false, even with an arbitrarily large positive zeroth-order coefficient.
ER -