On the anisotropic critical $p$-Laplace equation: classification, decomposition, and stability results

Abstract: We investigate both qualitative and quantitative issues related to the classification of non-negative energy solutions to the anisotropic critical $p$-Laplace equation in $\mathbb{R}<sup>n$, for $1<p<n$. Specifically, we establish an anisotropic version of Struwe's decomposition, along with the interaction estimate for the family of bubbles in this decomposition. Moreover, we provide a short proof of the classification result as well as a quantitative stability result, proving that every energy solution to a perturbation of the anisotropic critical equation must be closed to a bubble, in the absence of bubbling.

• The paper classifies all positive energy solutions as explicit (p,H)-bubbles under anisotropic conditions.
• It employs a refined Struwe-type decomposition to analyze bubble interactions and energy concentration in Palais–Smale sequences.
• The study establishes quantitative stability by linking energy deficits to the closeness of solutions to the optimal bubble profile.

Summary of "On the anisotropic critical $p$-Laplace equation: classification, decomposition, and stability results" (2604.13758)

Problem Formulation and Context

The paper studies the anisotropic critical $p$-Laplace equation: $$\Delta_p^H u + u^{p^*-1} = 0 \quad\text{in } \mathbb{R}^n,$$ where $1 < p < n$, $p^* = \frac{np}{n-p}$, and $\Delta_p^H$ is the so-called Finsler $p$-Laplacian defined via a general norm $H:\mathbb{R}^n\to [0,\infty)$. The system is tightly related to the sharp anisotropic Sobolev inequality: $$S_p \|u\|_{L^{p^*}(\mathbb{R}^n)} \leq \|H(\nabla u)\|_{L^p(\mathbb{R}^n)},$$ with equality attained by explicit $(p,H)$-bubbles.

The paper aims to generalize key results from the classical (Euclidean) critical $p$0-Laplace theory to the anisotropic setting, including:

• Classification of positive energy solutions.
• Decomposition phenomena for Palais–Smale sequences (Struwe-type results).
• Quantitative stability estimates, i.e., closeness to a bubble for perturbed equations under energy constraints.
• Analysis and estimates of bubble interactions in decompositions.

Classification of Positive Energy Solutions

The main classification result states that for any positive energy solution $p$1 to the anisotropic critical $p$2-Laplace equation and under suitable regularity and ellipticity assumptions on $p$3, $p$4 must be a $p$5-bubble: $p$6 with $p$7 given explicitly in terms of $p$8, the dual norm, and scaling/translation parameters.

The proof leverages integral estimates involving the so-called $p$9-function, which encode the structure of optimal Sobolev constants for the anisotropic norm and provide strong rigidity results. A concise argument circumvents technical regularity bottlenecks typical for general norms.

Struwe's Decomposition: Bubbling and Interaction

Extending Struwe’s global compactness theorem to the anisotropic context, the paper establishes that any bounded Palais–Smale sequence $$\Delta_p^H u + u^{p^*-1} = 0 \quad\text{in } \mathbb{R}^n,$$0 for the energy functional associated with the equation admits a decomposition

$$\Delta_p^H u + u^{p^*-1} = 0 \quad\text{in } \mathbb{R}^n,$$1

where each $$\Delta_p^H u + u^{p^*-1} = 0 \quad\text{in } \mathbb{R}^n,$$2 is a $$\Delta_p^H u + u^{p^*-1} = 0 \quad\text{in } \mathbb{R}^n,$$3-bubble, and the $$\Delta_p^H u + u^{p^*-1} = 0 \quad\text{in } \mathbb{R}^n,$$4 bubbles are widely separated and/or strongly scaled relative to each other. Strong interaction estimates are provided: for $$\Delta_p^H u + u^{p^*-1} = 0 \quad\text{in } \mathbb{R}^n,$$5, either their scales diverge or their centers become infinitely separated under suitable normalization: $$\Delta_p^H u + u^{p^*-1} = 0 \quad\text{in } \mathbb{R}^n,$$6 This precise control is new even for isotropic $$\Delta_p^H u + u^{p^*-1} = 0 \quad\text{in } \mathbb{R}^n,$$7-Laplace equations ($$\Delta_p^H u + u^{p^*-1} = 0 \quad\text{in } \mathbb{R}^n,$$8), and is expected to play a central role in further quantitative analyses of bubbling.

Quantitative Stability: Closeness in the Absence of Bubbling

The paper obtains a quantitative stability result for energy solutions to perturbed equations: $$\Delta_p^H u + u^{p^*-1} = 0 \quad\text{in } \mathbb{R}^n,$$9 assuming the energy is close to that of a single bubble. The deficit functional

$1 < p < n$0

measures how much the solution deviates from the ideal case. The main theorem proves that under an appropriate energy constraint and $1 < p < n$1, there exist $1 < p < n$2 and $1 < p < n$3 such that: $1 < p < n$4 for some $1 < p < n$5-bubble $1 < p < n$6. The constants depend only on structural parameters.

The proof adapts a scheme based on the $1 < p < n$7-function, as developed for the isotropic $1 < p < n$8-Laplace equation in [cg-plap], but incorporates the complications of anisotropy through careful tensor and integral estimates.

Technical Innovations and Integral Tensor Estimates

• The approach constructs a $1 < p < n$9-function whose integral estimates are leveraged to deduce strong rigidity—the solution's underlying geometry forces it toward the bubble profile.
• The paper establishes sharp bounds on the traceless part of the Hessian (associated with the stress field $p^* = \frac{np}{n-p}$0), quantifying deviation from the optimal structure.
• Regularity assumptions on the norm ($p^* = \frac{np}{n-p}$1, uniform convexity) are mainly technical; similar results likely hold under milder smoothness.
• The integral estimates are robust against anisotropy and do not rely on the Hilbert structure exploited in the classical ($p^* = \frac{np}{n-p}$2) case.

Open Problems and Future Directions

The paper discusses two major unresolved issues:

• Classification for local weak solutions: For lower $p^* = \frac{np}{n-p}$3 or in high dimensions, does there exist a positive weak solution not of bubble form? If so, understanding the precise threshold becomes a significant theoretical problem.
• Stability allowing bubbling: While complete quantitative stability with bubbling is known for $p^* = \frac{np}{n-p}$4, it remains an open technical challenge for general $p^* = \frac{np}{n-p}$5 and anisotropic norms.

Future research will likely focus on resolving these questions, generalizing stability results to broader function spaces and perturbations, and applying the anisotropic theory to geometric PDEs and nonlinear analysis.

Conclusion

This paper provides a comprehensive extension of the critical $p^* = \frac{np}{n-p}$6-Laplace theory to anisotropic frameworks, achieving sharp classification, decomposition, and stability results with new quantitative estimates. The methodological innovations in employing the $p^* = \frac{np}{n-p}$7-function and handling the anisotropic tensor structure are expected to influence further studies of quasilinear elliptic equations and geometric variational problems. The results support both theoretical and practical advances in nonlinear PDE, particularly for variational theories associated with anisotropic Sobolev inequalities.