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On the stability of the critical $p$-Laplace equation

Giulio Ciraolo, Michele Gatti

TL;DR

This work addresses the stability of perturbations of the critical $p$-Laplacian in $\mathbb{R}^n$ for $1<p<n$ under a non-bubbling energy regime. The authors develop a PDE-centered quantitative framework based on a $P$-function and associated stress tensor to show that any positive weak solution to a perturbed equation, with energy near that of a single bubble and satisfying a normalization $\kappa_0(u)=1$, is close in $\mathcal{D}^{1,p}(\mathbb{R}^n)$ to a $p$-bubble, with a bound controlled by the Sobolev-deficit $\mathrm{def}(u,\kappa)$ raised to a power $\vartheta\in(0,1)$. The proof proceeds via Struwe-type reduction to a single bubble, sharp decay estimates, a differential identity for the $P$-function, a regularized approximation scheme, and a careful construction of a $p$-paraboloid-based approximation that inverts to a bubble; the strategy culminates in transferring these estimates back to $u$ and undoing the symmetry reductions. The paper also discusses optimality, notes an upcoming sharp-exponent result by Liu & Zhang, and applies the stability framework to a quasi-symmetry result, illustrating practical implications for symmetry properties of solutions. Overall, the work extends quantitative bubbling stability from the linear case $p=2$ to the nonlinear $p$-Laplacian setting and provides tools potentially adaptable to anisotropic contexts.

Abstract

For $1<p<n$, it is well-known that non-negative, energy weak solutions to $Δ_p u + u^{p^{\ast}-1} =0$ in $\mathbb{R}^n$ are completely classified. Moreover, due to a fundamental result by Struwe and its extensions, this classification is stable up to bubbling. In the present work, we investigate the stability of perturbations of the critical $p$-Laplace equation for any $1<p<n$, under a condition that prevents bubbling. In particular, we show that any solution $u \in \mathcal{D}^{1,p}(\mathbb{R}^n)$ to such a perturbed equation must be quantitatively close to a bubble. This result generalizes a recent work by the first author, together with Figalli and Maggi (Int. Math. Res. Not. IMRN 2018 (2018), no. 21, 6780-6797), in which a sharp quantitative estimate was established for $p=2$. However, our analysis differs completely from theirs and is based on a quantitative $P$-function approach.

On the stability of the critical $p$-Laplace equation

TL;DR

This work addresses the stability of perturbations of the critical -Laplacian in for under a non-bubbling energy regime. The authors develop a PDE-centered quantitative framework based on a -function and associated stress tensor to show that any positive weak solution to a perturbed equation, with energy near that of a single bubble and satisfying a normalization , is close in to a -bubble, with a bound controlled by the Sobolev-deficit raised to a power . The proof proceeds via Struwe-type reduction to a single bubble, sharp decay estimates, a differential identity for the -function, a regularized approximation scheme, and a careful construction of a -paraboloid-based approximation that inverts to a bubble; the strategy culminates in transferring these estimates back to and undoing the symmetry reductions. The paper also discusses optimality, notes an upcoming sharp-exponent result by Liu & Zhang, and applies the stability framework to a quasi-symmetry result, illustrating practical implications for symmetry properties of solutions. Overall, the work extends quantitative bubbling stability from the linear case to the nonlinear -Laplacian setting and provides tools potentially adaptable to anisotropic contexts.

Abstract

For , it is well-known that non-negative, energy weak solutions to in are completely classified. Moreover, due to a fundamental result by Struwe and its extensions, this classification is stable up to bubbling. In the present work, we investigate the stability of perturbations of the critical -Laplace equation for any , under a condition that prevents bubbling. In particular, we show that any solution to such a perturbed equation must be quantitatively close to a bubble. This result generalizes a recent work by the first author, together with Figalli and Maggi (Int. Math. Res. Not. IMRN 2018 (2018), no. 21, 6780-6797), in which a sharp quantitative estimate was established for . However, our analysis differs completely from theirs and is based on a quantitative -function approach.

Paper Structure

This paper contains 18 sections, 9 theorems, 341 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline of the proof.
  4. Structure of the paper.
  5. Symmetries of the problem and proof of Corollary \ref{['cor:main-bubbles']}
  6. Proof of Theorem \ref{['th:main-bubbles']}
  7. Application of the Struwe-type result and reduction.
  8. Sharp decay estimates in quantitative form.
  9. Introduction of the $P$-function and of the relevant vector fields.
  10. A differential identity.
  11. The approximation procedure.
  12. Quantitative integral estimate.
  13. Fundamental estimate for $W$.
  14. Construction of the approximating functions.
  15. Going back to $u$.
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $n \in \mathbb{N}$, $1<p<n$, and $\kappa \in L^\infty(\mathbb{R}^n) \cap C^{1,1}_{{\rm loc}}(\mathbb{R}^n)$ a positive function. Let $u \in \mathcal{D}^{1,p}(\mathbb{R}^n)$ be a positive weak solution to eq:maineq-bubb satisfying eq:ipotesi-energ. Moreover, suppose that where $\kappa_0(u)$ is defined in eq:defk0. Then, there exist a large constant $C \geq 1$, a small $\vartheta \in (0,1)$, an

Figures (1)

  • Figure 1: The functions considered and the region $B_r$ where the energy is concentrated. The dotted blue line is $1/P$ up to the correct factor.

Theorems & Definitions (15)

  • Theorem 1.1
  • Corollary 1.1.1
  • Lemma 2.1
  • proof : Proof of Corollary \ref{['cor:main-bubbles']}
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • Lemma 3.1
  • ...and 5 more