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Sharp homogeneous Gagliardo--Nirenberg inequalities and normalized solutions for generalized stationary MMT equations
Abstract
We prove the existence of optimizers for a class of homogeneous Gagliardo-Nirenberg inequalities of the form |Dsϕ|Lq ≤ C |Ds1ϕ|Lp1−θ |Ds2ϕ|L2θ, s1<s2, under the scaling relation 1/q−s/d = (1−θ)(1/p−s1/d) + θ(1/2−s2/d), and the strict interpolation condition s<(1−θ)s1+θs2. When s>s1, the problem reduces, after shifting the derivative orders, to the result of Bellazzini, Frank and Visciglia, whereas the case s≤ s1 follows from the present paper. As an application, we study normalized solutions of the associated Euler-Lagrange equations under the constraint |Ds1ψ|Lpp=λ>0. In particular, the case p=2 includes the stationary equation arising from the Majda-McLaughlin-Tabak (MMT) model.
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BibTeX
@misc{esfahani2026sharp,
title = {{Sharp homogeneous Gagliardo--Nirenberg inequalities and normalized solutions for generalized stationary MMT equations}},
author = {Amin Esfahani and Mukhtar Karazym},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.08686},
eprint = {2608.08686},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.08686},
}
RIS
TY - UNPB
TI - Sharp homogeneous Gagliardo--Nirenberg inequalities and normalized solutions for generalized stationary MMT equations
AU - Amin Esfahani
AU - Mukhtar Karazym
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2608.08686
UR - https://arxiv.org/abs/2608.08686
AB - We prove the existence of optimizers for a class of homogeneous Gagliardo-Nirenberg inequalities of the form |D^sϕ|_L^q ≤ C |D^s_1ϕ|_L^p^1−θ |D^s_2ϕ|_L^2^θ, s_1<s_2, under the scaling relation 1/q−s/d = (1−θ)(1/p−s_1/d) + θ(1/2−s_2/d), and the strict interpolation condition s<(1−θ)s_1+θs_2. When s>s_1, the problem reduces, after shifting the derivative orders, to the result of Bellazzini, Frank and Visciglia, whereas the case s≤ s_1 follows from the present paper. As an application, we study normalized solutions of the associated Euler-Lagrange equations under the constraint |D^s_1ψ|_L^p^p=λ>0. In particular, the case p=2 includes the stationary equation arising from the Majda-McLaughlin-Tabak (MMT) model.
ER -