A generalisation of the Gagliardo--Nirenberg Inequality with applications to mass-critical and mass-subcritical elliptic equations

Via a new inequality à la Gagliardo–Nirenberg, we prove the existence and nonexistence of solutions to \begin{equation*} \begin{cases} (-Δ)^s u + \fracμ{|y|^{2s}} u + λu = f(u), \quad $\mathbb{R}$^N \ni x = (y,z) \in $\mathbb{R}$^K \times $\mathbb{R}$^{N-K}, \ \int_{$\mathbb{R}$^N} u^2 , $\mathrm{d}$x = ρ\end{cases} \end{equation*} in the mass-critical and mass-subcritical regimes, where $s>0$, $N\ge K\ge 2$, $\mu \in \mathbb{R}$ belongs to a specific range, $\rho >0$ is given a priori, and $\lambda \in \mathbb{R}$ is unknown. Additionally, we obtain similar results for the problem above with $\mu =0$ and $N\ge 1$ as well as a related curl-curl equation. Finally, we provide a thorough insight into the threshold for $\rho$ that divides the scenarios of negative and zero least energy.