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 $$\Vert D^s\varphi {\Vert }_{L^q}\le C\Vert D^{s_1}\varphi {\Vert }_{L^p}^{1-\theta }\Vert D^{s_2}\varphi {\Vert }_{L^2}^{\theta },s_1<s_2,$$ under the scaling relation $$\frac{1}{q}-\frac{s}{d}=(1-\theta )\left(\frac{1}{p}-\frac{s_1}{d}\right)+\theta \left(\frac{1}{2}-\frac{s_2}{d}\right),$$ and the strict interpolation condition $$s<(1-\theta )s_1+\theta s_2.$$ When $s>s_1$, the problem reduces, after shifting the derivative orders, to the result of Bellazzini, Frank and Visciglia, whereas the case $s\le s_1$ follows from the present paper. As an application, we study normalized solutions of the associated Euler-Lagrange equations under the constraint $$\Vert D^{s_1}\psi {\Vert }_{L^p}^p=\lambda >0.$$ In particular, the case $p=2$ includes the stationary equation arising from the Majda-McLaughlin-Tabak (MMT) model.