Critical $p$-Laplace equations with monotone coefficients: Liouville classification and a Schoen-type Harnack inequality

Technical Summary: Critical p-Laplace Equations with Monotone Coefficients

Problem Statement
This paper investigates positive weak solutions to the critical $p$-Laplace equation with a non-autonomous nonlinearity:
$$-\Delta_p u = u^{p^*-1}h(u), \quad \text{in } \mathbb{R}^n,$$
where $1 < p < n$, $p^* = \frac{np}{n-p}$, and $h$ is a positive, bounded, continuous, and nonincreasing function. The study focuses on two interconnected objectives:

1. Liouville Classification: Determining the precise form of positive entire solutions without imposing a priori decay assumptions (e.g., finite energy or specific growth rates).
2. Scale-Invariant Harnack Inequality: Establishing a local estimate of the form $\left(\sup_{B_R} u\right) \left(\inf_{B_{2R}} u\right)^{p-1} \leq C R^{p-n}$ for nonnegative weak solutions defined in $B_{3R}$.

The work addresses a significant gap in the theory of quasilinear equations. Unlike the semilinear case ($p=2$), the $p$-Laplacian ($p \neq 2$) lacks a Kelvin transform that preserves the equation, rendering classical moving plane/sphere methods inapplicable in their standard form. Furthermore, existing classification results for the purely critical equation ($h \equiv \text{const}$) often rely on restrictive assumptions (finite energy, boundedness) or specific ranges of $p$. This paper aims to provide a classification-to-Harnack theory for the full range $1 < p < n$ under a monotonicity assumption on $h$.

Methodology
The author develops a framework that serves as a quasilinear counterpart to the Kelvin transform and the method of moving spheres, adapted for the $p$-Laplacian with continuous monotone coefficients. The methodology proceeds in two main stages:

1. Global Classification via Invariant Tensors and Weak Bochner Identities:

   • Transformation: The solution $V$ is transformed via $w = V^{-1/a}$ (where $a = \frac{n-p}{p}$), converting the equation into a form amenable to stress-tensor analysis.
   • Stress Tensor and P-function: The author defines a nonlinear stress field $X_w = |\nabla w|^{p-2}\nabla w$ and a modified P-function $Q_w$.
   • Handling Monotonicity: A key technical innovation is the decomposition of the transformed coefficient $c_\tau(t)$ into an absolutely continuous part and a singular (Stieltjes) defect $e_\tau(t)$. This allows the author to handle the fact that $h$ is only continuous and monotone, potentially containing a Cantor part in its derivative.
   • Weak Bochner Identity: A new weak Bochner-type identity is derived for $Q_w$. Unlike previous approaches that relied on global integral estimates (e.g., Ou [34]), this work utilizes a local pointwise coercive term.
   • Multiplier Construction: To control the growth of $Q_w$ and prevent positive maxima from escaping to infinity, the author constructs a specific weight function $\phi_{\ell,m}$ by solving a backward ordinary differential equation tailored to the target PDE inequality. This replaces the fixed power weights used in the pure-power case.
   • Rigidity Argument: By establishing a sharp global bound $Q_w \leq c_{\tau,0}$ and analyzing the behavior at minimum points (using tangent analysis and punctured-ball barriers), the author proves that the solution must coincide with an Aubin–Talenti profile. Crucially, they show that the existence of such a profile forces the coefficient $h$ to be constant on the interval of values attained by the profile.

2. Harnack Inequality via De-concentration and Potential Theory:

   • Contradiction Argument: The proof of the Harnack inequality proceeds by contradiction, assuming a sequence of solutions violates the estimate.
   • Blow-up Analysis: A rescaling procedure identifies a limiting entire solution. The classification result (Theorem 1.1) is then applied to this limit.
   • First-Contact Mechanism: Instead of relying on a preliminary "neck estimate" (as in Qin–Zhang [35]), the author employs a "first-contact" argument combined with potential-theoretic de-concentration.
   • Kilpeläinen–Malý Estimates: Using the theory of nonlinear potentials (specifically Wolff potentials), the author derives a uniform bound on the source term near the contact point. This establishes the necessary coefficient control to apply the Harnack inequality, effectively removing the need for a priori neck estimates.

Key Contributions and Results

• Theorem 1.1 (Classification of Normalized Profiles): For any $\tau \in [0, \infty]$, any normalized, bounded, positive entire solution to the equation with the scaled nonlinearity $g_\tau$ must be an Aubin–Talenti profile:
  $$V(x) = \left(1 + \gamma_\tau |x|^{p'}\right)^{-a}.$$
  Furthermore, if $0 < \tau < \infty$, the existence of such a solution implies that $h(s) = h(\tau)$ for all $0 < s \leq \tau$. This establishes a rigidity theorem for both the solution and the coefficient.

• Theorem 1.2 (Scale-Invariant Harnack Inequality): For nonnegative weak solutions in $B_{3R}$, the following estimate holds:
  $$\left(\sup_{B_R} u\right) \left(\inf_{B_{2R}} u\right)^{p-1} \leq C R^{p-n}.$$
  This result is established without assuming finite energy or specific asymptotic behavior at infinity.

• Corollary 1.4 (Unrestricted Liouville Theorem): Every positive entire solution to the original equation is an Aubin–Talenti profile (up to scaling and translation), and the coefficient $h$ is constant on the range of the solution.

• Equivalence of Properties: The paper demonstrates that for the purely critical equation, the unrestricted Liouville classification is equivalent to the Schoen-type Harnack inequality.

Significance and Claims
The author claims that their arguments constitute the first quasilinear counterparts of the Kelvin transform and the method of moving spheres that are available for general $1 < p < n$ with merely continuous monotone coefficients.

• Methodological Advancement: The work provides alternative proofs for classical results in the semilinear case ($p=2$) and extends them to the quasilinear setting where standard conformal invariance tools fail. The introduction of the weak Bochner identity with a tailored multiplier and the handling of singular Stieltjes measures in the coefficient are presented as novel analytical tools.
• Removal of Assumptions: The results provide a "fully unrestricted" Liouville theorem, removing the need for finite energy, boundedness, or prescribed growth conditions that characterized previous classification results for $p \neq 2$.
• Structural Insight: The paper highlights that the interaction between Liouville theorems and Harnack inequalities is fundamental. The classification of blow-up profiles is shown to be indispensable for establishing the Harnack estimate when the nonlinearity is not purely power-type.

The paper does not propose new applications or experimental proposals but positions itself as a foundational contribution to the compactness theory for critical elliptic equations, bridging the gap between the well-understood semilinear case and the more rigid quasilinear setting.