Degenerate stability of critical points of the Caffarelli-Kohn-Nirenberg inequality along the Felli-Schneider curve

Problem Statement
This paper investigates the quantitative stability of the critical Hardy-Hénon equation, which serves as the Euler-Lagrange equation for the classical Caffarelli-Kohn-Nirenberg (CKN) inequality. Specifically, the authors focus on the regime where the parameters $(a, b)$ lie exactly on the Felli-Schneider curve ($b = b_{FS}(a)$ with $a < 0$).

In this setting, the equation is known to be degenerate. Unlike the non-degenerate cases (where $a \ge 0$ or $b > b_{FS}(a)$), the linearized operator around a Talenti bubble (the positive solution) possesses a non-trivial kernel beyond the standard scaling and translation invariances. Previous work by Wei and Wu established optimal quantitative stability estimates for the non-degenerate case, relying heavily on the non-degeneracy condition. However, these results fail on the Felli-Schneider curve because the degeneracy prevents the standard linear estimates from holding. The central problem addressed is whether a suitable quantitative stability estimate exists in this degenerate setting and, if so, what the optimal form of the stability function $F$ is.

Methodology
The authors employ a multi-step analytical approach:

1. Emden-Fowler Transformation: The problem is reformulated from the Euclidean space $\mathbb{R}^n$ to the cylinder $\mathcal{C} = \mathbb{R} \times S^{n-1}$. This transformation converts the weighted CKN inequality and the Hardy-Hénon equation into a Gagliardo-Nirenberg interpolation inequality and a corresponding semilinear elliptic equation with constant coefficients on the cylinder. This simplifies the spectral analysis and the handling of the weight functions.
2. Local-to-Global Strategy: Following the global compactness principle, the authors reduce the global stability problem to a local one. They prove that if a function is sufficiently close to a sum of Talenti bubbles, the stability estimate holds locally; the global result then follows by combining this with the compactness principle.
3. High-Order Expansion Techniques: To handle the degeneracy, the authors cannot rely on second-order expansions (sufficient in non-degenerate cases). Instead, they utilize high-order expansion techniques (up to the fourth order) for the nonlinear terms. This involves splitting the integration domain and carefully estimating the remainder terms using specific inequalities for the function $|x+y|^{p-2}(x+y)$.
4. Spectral Analysis and Orthogonal Projections: The analysis involves decomposing the perturbation of the solution into components parallel and orthogonal to the kernel of the linearized operator (denoted as $X$ or $Y$ in the cylinder setting). The authors derive estimates for the orthogonal component and analyze the interaction between the degenerate directions and the nonlinear terms.
5. Construction of Counterexamples/Optimality: To prove the sharpness of their estimates, the authors construct specific sequences of functions. This construction is technical; it involves perturbing natural candidates with solutions to special Schrödinger equations to achieve the desired cubic (or higher) scaling, rather than the quadratic scaling that simpler perturbations would yield.

Key Contributions and Results

• Optimal Stability for Single Bubble ($\nu=1$): For the case of a single bubble ($\nu=1$), the authors establish a sharp cubic stability estimate:
  $$\inf_{\lambda > 0} \|u - U_\lambda\|_{D^{1,2}_a}^3 \le C \|H(u)\|_{D^{-1,2}_a}$$
  They prove this estimate is optimal by constructing a sequence where the ratio of the error to the cubic power of the distance remains bounded. Furthermore, they provide a refined estimate for the orthogonal component of the solution, showing that if the solution is "not too degenerate" (satisfying a specific condition relating the orthogonal and degenerate components), a linear estimate holds for the orthogonal part.

• Stability for Multiple Bubbles ($\nu \ge 2$): For the case of multiple bubbles ($\nu \ge 2$), the authors derive a stability estimate for the orthogonal component of the solution, provided the solution is not "too degenerate." The estimate takes the form:
  $$\|\Pi^\perp_X(u) - \sum U_{\lambda_i}\| \le C F_1(\|H(u)\|)$$
  where $F_1$ depends on the dimension and the exponent $p$. The condition "not too degenerate" is necessary, as the authors demonstrate via a counterexample that without it, the stability estimate fails.

• Sharpness and Optimality: The paper explicitly addresses the optimality of the derived exponents. For $\nu=1$, the cubic estimate is shown to be sharp. For $\nu \ge 2$, the authors establish the validity of the estimate under the non-degeneracy restriction and show that this restriction is optimal in certain regimes ($p \ge 3$ or $\nu \ge 2, p > 3$).

Significance
The authors claim that this work provides the first instance of degenerate stability results in the critical point setting for the Caffarelli-Kohn-Nirenberg inequality. Prior to this, quantitative stability results were largely dependent on non-degeneracy conditions. By successfully navigating the degeneracy on the Felli-Schneider curve, the paper extends the theory of quantitative stability to a critical boundary case where previous methods failed. The authors suggest that their methods, particularly the high-order expansion and the specific construction of test functions and perturbation sequences, may be applicable to other problems involving degenerate stability.

The paper concludes that while the stability in the functional setting (inequality stability) has been studied extensively, the critical point setting on the Felli-Schneider curve presents unique challenges due to the lack of non-degeneracy, which this work resolves for the single bubble case and partially for the multi-bubble case under specific conditions.