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

This paper, titled "A Generalisation of the Gagliardo–Nirenberg Inequality with Applications to Mass-Critical and Mass-Subcritical Elliptic Equations" by Bartosz Bieganowski and Jacopo Schino, presents a significant advancement in the variational analysis of fractional and polyharmonic elliptic equations involving Hardy-type potentials.

Below is a detailed technical summary of the work.

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1. The Problem

The authors investigate the existence and nonexistence of solutions to a class of nonlinear elliptic equations with a prescribed $L^2$-norm (mass) constraint. The primary equation is:
$$(-\Delta)^s u + \frac{\mu}{|y|^{2s}} u + \lambda u = f(u), \quad x = (y, z) \in \mathbb{R}^K \times \mathbb{R}^{N-K}$$
subject to the constraint:
$$\int_{\mathbb{R}^N} u^2 \, dx = \rho$$
where:

• $s > 0$: Represents the order of the operator (fractional or polyharmonic).
• $\mu \in \mathbb{R}$: A parameter for the singular Hardy-type potential.
• $\lambda \in \mathbb{R}$: An unknown Lagrange multiplier arising from the mass constraint.
• $f(u)$: A nonlinearity satisfying specific growth conditions.
• $\rho > 0$: A prescribed mass.

The problem is categorized into mass-subcritical and mass-critical regimes based on the growth of the nonlinearity $f$ relative to the exponent $2^\# = 2 + \frac{4s}{N}$.

2. Methodology

The authors employ variational methods, specifically looking for minimizers of the energy functional:
$$J(u) = \frac{1}{2} \int_{\mathbb{R}^N} \left( |D^s u|^2 + \frac{\mu}{|y|^{2s}} u^2 \right) dx - \int_{\mathbb{R}^N} F(u) \, dx$$
within the symmetric subspace $X_\mu^G$ (functions invariant under a specific group of rotations).

Key methodological steps include:

• New Gagliardo–Nirenberg Inequality: The core of the paper is the derivation of a generalized Gagliardo–Nirenberg inequality that incorporates the singular potential $\mu|y|^{-2s}$. This inequality provides the optimal constant $C_{N,p}$ used to control the nonlinearity by the energy functional.
• Concentration-Compactness: To handle the loss of compactness due to the unboundedness of $\mathbb{R}^N$, the authors use a version of Lions' Lemma adapted to the symmetry of the problem (specifically handling translations in the $z$-variable).
• Constraint Analysis: They define a threshold mass $\rho^*$ that separates the regime where the minimum energy $m(\rho)$ is zero from the regime where it is negative.

3. Key Contributions and Results

A. Existence and Nonexistence of Solutions

• Theorem 1.1 (Existence): Provides conditions under which global minimizers exist for the mass-subcritical regime. It establishes upper bounds for the threshold mass $\rho^*$.
• Theorem 1.4 (Local Minimizers): In the mass-critical regime, the authors prove the existence of local minimizers (solutions with positive energy) when the mass $\rho$ lies in a specific interval $(\rho^*, \rho^{**})$.
• Theorem 1.6 (Nonexistence): A novel result stating that if the nonlinearity $f$ satisfies certain strict monotonicity conditions relative to the power $2^\#$, no nontrivial global minimizers exist at the threshold mass $\rho^*$.

B. The Threshold Mass $\rho^*$

The paper provides a thorough investigation into the threshold $\rho^*$. They show that $\rho^*$ is the value that divides scenarios of negative energy (where solutions exist) and zero energy. They provide explicit bounds and discuss how the behavior of $F(t)$ at the origin and infinity dictates this threshold.

C. Extension to Curl-Curl Equations

The authors demonstrate the utility of their framework by applying it to cylindrically symmetric curl-curl equations in $\mathbb{R}^3$. By using a specific ansatz, they show that the vector problem reduces to the scalar problem studied in the paper, allowing them to export their existence/nonexistence results to electromagnetic modeling.

4. Significance

The significance of this work lies in three areas:

1. Mathematical Generality: It unifies several disparate problems (fractional Schrödinger, polyharmonic, and Hardy-type equations) under a single variational framework.
2. Sharpness: By providing a new Gagliardo–Nirenberg inequality, the authors offer sharper tools for studying the stability and existence of standing waves in quantum mechanics and nonlinear optics.
3. Completeness: Unlike previous literature that often focuses solely on the $\mu=0$ case or specific $s=1$ cases, this paper provides a comprehensive treatment of the $\mu \neq 0$ case across various dimensions and operator orders, including the critical mass threshold which is often the most difficult to characterize.