Sharp homogeneous Gagliardo--Nirenberg inequalities and normalized solutions for generalized stationary MMT equations

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)