发表机构
Nazarbayev University; Astana IT University（纳扎尔巴耶夫大学; 阿斯塔纳IT大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明尖锐齐次Gagliardo–Nirenberg不等式的最优解存在性，扩展了相关已有结果，并将其应用于证明广义MMT型方程的归一化解存在性。
AI 中文摘要
本文证明了一类尖锐齐次Gagliardo–Nirenberg不等式的最优解存在性，扩展了Bellazzini、Frank和Visciglia的结果。作为应用，建立了对应欧拉-拉格朗日方程的归一化解存在性，其中p=2的情形包含来自Majda–McLaughlin–Tabak（MMT）模型的定态方程。
英文摘要
We prove the existence of optimizers for a class of homogeneous Gagliardo-Nirenberg inequalities of the form $$ \|D^{s}ϕ\|_{L^q} \leq C \|D^{s_1}ϕ\|_{L^p}^{1-θ} \|D^{s_2}ϕ\|_{L^2}^θ, \quad s_1<s_2, $$ under the scaling relation $$ \frac{1}{q}-\frac{s}{d} = (1-θ)\left(\frac{1}{p}-\frac{s_1}{d}\right) + θ\left(\frac{1}{2}-\frac{s_2}{d}\right), $$ 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 $s\leq s_1$ follows from the present paper. As an application, we study normalized solutions of the associated Euler-Lagrange equations under the constraint $$ \|D^{s_1}ψ\|_{L^p}^p=λ>0. $$ In particular, the case $p=2$ includes the stationary equation arising from the Majda-McLaughlin-Tabak (MMT) model.
