A Pohozaev-type neck proof of a conditional Harnack inequality in the critical $p$-Laplacian setting

Technical Summary: A Pohozaev-Type Neck Proof of a Conditional Harnack Inequality in the Critical $p$-Laplacian Setting

Problem Statement
The paper addresses the regularity and quantitative behavior of positive weak solutions to the critical quasilinear equation:
$$-\Delta_p u = g(u) \quad \text{in } \Omega \subset \mathbb{R}^n,$$
where $1 < p < n$, $\Delta_p u = \text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian, and the nonlinearity $g$ satisfies critical Sobolev growth conditions (specifically $g(s) \sim s^{p^*-1}$ as $s \to \infty$, where $p^* = \frac{np}{n-p}$).

The central objective is to establish a Harnack inequality of the form:
$$\left(\sup_{B_R} u\right) \left(\inf_{B_{2R}} u\right)^{p-1} \leq C R^{p-n},$$
which provides a pointwise description of singular solutions and underpins blow-up analysis.

Methodological Context and Obstacles
In the semilinear case ($p=2$), such inequalities were established by Schoen and later extended by Li and Zhang using the moving sphere (or moving plane) method. This method relies heavily on the Kelvin transform, which exploits the conformal invariance of the Laplacian.

However, for $p \neq 2$, the $p$-Laplacian is not conformally invariant, rendering the Kelvin transform and the associated moving sphere method unavailable. This absence constitutes a genuine obstruction to extending classical classification and Harnack results to the general critical $p$-Laplacian setting.

Methodology: The Pohozaev-Neck Argument
To overcome the lack of conformal invariance, the authors develop a Pohozaev-neck argument. This approach replaces the geometric reflection techniques of the $p=2$ case with an analytic argument based on the Pohozaev identity and the structure of isolated singularities. The methodology proceeds through the following steps:

1. Structural Sign Condition: Under the assumption that $s \mapsto s^{-(p^*-1)}g(s)$ is nonincreasing, the authors establish that the Pohozaev functional is non-negative on balls for regular solutions.
2. Singularity Analysis: They utilize the Kichenassamy–Véron pole theorem to analyze the behavior of solutions near isolated singularities. They demonstrate that if a solution has a "regular part" (a non-zero constant term $\beta$) in its asymptotic expansion $u(x) \sim \kappa|x|^{-\alpha} + \beta$, the Pohozaev functional at the singularity becomes strictly negative.
3. The Neck Lemma (Lemma 3.7): This is the core technical contribution. The lemma assumes a "preliminary" singular decay rate (specifically $|x|^{-(n-p)/p}$) on a normalized neck region. By rescaling the solution to a "neck" scale, the authors construct a limit profile.
   • If the decay were slower than the sharp rate, the limit profile would possess a positive regular part ($\beta > 0$).
   • This would imply a negative Pohozaev value for the limit.
   • However, the Pohozaev sign condition inherited from the original equation (via the monotonicity of $g$) requires the value to be non-negative.
   • This contradiction forces the regular part to vanish, thereby upgrading the preliminary decay rate to the sharp p-harmonic fundamental rate $|x|^{-(n-p)/(p-1)}$.

Key Assumptions (Conditional Nature)
The main result is conditional, relying on two explicit hypotheses:

1. Entire Classification Hypothesis: Every bounded positive entire weak solution to the limiting equation $-\Delta_p V = g_A(V)$ in $\mathbb{R}^n$ (where $g_A$ represents the blow-up limit of the nonlinearity) must be an Aubin–Talenti $p$-bubble.
2. Preliminary Neck Control: There exists a preliminary upper bound on the normalized neck solutions of the form $v_j(y) \leq C|y|^{-(n-p)/p}$.

Main Results

• Theorem 1.3 (Harnack Inequality): Under the structural assumptions on $g$ (Assumption 1.1) and the two hypotheses above (Assumption 1.2), the conditional Harnack inequality holds for any positive weak solution in $B_{3R}$.
• Sharp Decay Upgrade: The paper proves that the Pohozaev-neck argument successfully upgrades the preliminary singular decay rate $|x|^{-(n-p)/p}$ to the sharp rate $|x|^{-(n-p)/(p-1)}$, which is necessary for the sup $\times$ inf estimate.
• Unconditional Cases: The authors note that in the pure-power case $g(s) = L s^{p^*-1}$, the classification hypothesis is known to hold for specific ranges of $n$ and $p$ (e.g., $n \leq 6$ or $n \geq 7$ with $p > n/3$), making the theorem unconditional in those regimes. Additionally, if a scale-invariant critical energy bound is assumed, the classification and preliminary control follow from existing finite-energy theory, rendering the result unconditional in that setting.

Significance and Claims
The paper claims to provide the first proof of a conditional Harnack inequality for the critical $p$-Laplacian ($p \neq 2$) that does not rely on conformal invariance.

• Methodological Shift: It introduces a Pohozaev-based "neck lemma" as a viable alternative to the Kelvin transform for analyzing bubbling phenomena in non-conformally invariant settings.
• Bridging the Gap: The work connects the local blow-up analysis (neck behavior) with global classification results. It explicitly frames the difficulty of the general case as the potential existence of bounded entire solutions that are not Aubin–Talenti bubbles (infinite-energy limits), a scenario excluded by the conditional hypothesis.
• Limitations: The authors are modest about the scope, acknowledging that without the classification hypothesis (or an energy bound), the existence of non-bubble entire solutions remains an open obstruction. The result is presented as a conditional step forward, dependent on the validity of the classification of entire blow-up limits.

Conclusion
The paper establishes a robust framework for analyzing critical $p$-Laplacian equations by replacing geometric conformal methods with analytic Pohozaev identities. It successfully derives a sharp Harnack inequality under the assumption that the classification of entire solutions behaves similarly to the semilinear case, thereby isolating the classification problem as the sole remaining hurdle for a fully unconditional result.