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

TL;DR

This paper introduces a new Gagliardo--Nirenberg type inequality to analyze the existence and nonexistence of solutions for certain fractional elliptic equations in critical and subcritical regimes, revealing thresholds for solution energy levels.

Contribution

It generalizes the Gagliardo--Nirenberg inequality and applies it to fractional elliptic equations with singular potentials, providing new existence results and energy thresholds.

Findings

Established existence and nonexistence results for fractional elliptic equations.

Identified thresholds for the mass parameter  that determine solution energy scenarios.

Extended results to related curl-curl 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.