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

Abstract

Via a new inequality \`a la Gagliardo--Nirenberg, we prove the existence and nonexistence of solutions to \begin{equation*} \begin{cases} (-\Delta)^s u + \frac{\mu}{|y|^{2s}} u + \lambda u = 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 = \rho \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.