Critical quasi-linear Schrödinger system with $p$-Laplacian

Technical Summary: Classification of Positive Solutions to a Critical Quasi-linear Schrödinger System with p-Laplacian

Problem Statement
This paper investigates the qualitative properties and classification of positive weak solutions $(u, v)$ to the $D^{1,p}(\mathbb{R}^N)$-critical quasi-linear Schrödinger system involving the $p$-Laplacian operator:
$$\begin{cases} -\Delta_p u = u^\alpha v^\beta & \text{in } \mathbb{R}^N, \\ -\Delta_p v = u^\beta v^\alpha & \text{in } \mathbb{R}^N, \end{cases}$$
where $1 < p < N$, $N \ge 2$, and the exponents satisfy $0 \le \alpha \le \beta$ with $\alpha + \beta = p^* - 1$. Here, $p^* = \frac{Np}{N-p}$ denotes the Sobolev critical exponent, and the solutions belong to the space $D^{1,p}(\mathbb{R}^N)$, defined as the completion of $C_0^\infty(\mathbb{R}^N)$ under the norm $\|\nabla u\|_{L^p}$.

The study addresses the generalization of uniqueness and classification results previously established for the semi-linear case ($p=2$) by Li and Ma [48] and Quittner and Souplet [58] to the fully quasi-linear regime $1 < p < N$.

Methodology and Technical Challenges
The authors employ the method of moving planes to establish radial symmetry and monotonicity. However, extending this method from $p=2$ to $1 < p < N$ presents significant analytical hurdles due to the non-linear nature of the $p$-Laplacian ($\Delta_p u = \text{div}(|\nabla u|^{p-2}\nabla u)$). The paper identifies four primary difficulties:

1. Non-linearity: Linear operations cannot be exchanged with $\Delta_p$, and standard comparison principles derived from maximum principles are not directly applicable.
2. Absence of Kelvin Transforms: The lack of conformal invariance for $p \neq 2$ prevents the use of Kelvin transforms, necessitating the derivation of sharp asymptotic estimates for both the solutions and their gradients.
3. Lack of Green's Representation: The absence of a Green integral representation formula precludes indirect methods based on equivalent integral equations.
4. Singular/Degenerate Ellipticity: The operator is singular elliptic for $1 < p < 2$ and degenerate elliptic for $p > 2$, requiring separate treatment of these cases as no uniform strong comparison principle exists.

To overcome these, the authors:

• Establish regularity and sharp asymptotic estimates for $(u, v)$ and $(|\nabla u|, |\nabla v|)$ without relying on Kelvin transforms.
• Utilize De Giorgi-Moser-Nash iteration techniques to derive preliminary decay bounds.
• Apply scaling arguments and doubling lemmas to refine these bounds into sharp estimates.
• Employ weighted Poincaré-type inequalities and Hardy-Sobolev inequalities to handle the moving plane procedure in the presence of critical sets (where gradients vanish).
• Discuss the singular ($1 < p < 2$) and degenerate ($p > 2$) cases separately, adapting the moving plane technique and comparison principles accordingly.

Key Contributions and Results

1. Regularity and Sharp Asymptotic Estimates (Theorem 1.3):
   The authors prove that any positive weak solution $(u, v)$ belongs to $C^{1,\theta}_{\text{loc}}(\mathbb{R}^N) \cap L^\infty(\mathbb{R}^N)$. Crucially, they establish sharp two-sided bounds for the solutions and their gradients for large $|x|$:
   $$\frac{c_0}{1 + |x|^{\frac{N-p}{p-1}}} \le u(x), v(x) \le \frac{C_0}{1 + |x|^{\frac{N-p}{p-1}}},$$
   $$\frac{c'_0}{|x|^{\frac{N-1}{p-1}}} \le |\nabla u(x)|, |\nabla v(x)| \le \frac{C'_0}{|x|^{\frac{N-1}{p-1}}}.$$
   These estimates are derived without Kelvin transforms, filling a gap in the literature for general $p$.

2. Radial Symmetry and Monotonicity (Theorem 1.4):
   Using the method of moving planes, the paper proves that all positive solutions are radially symmetric and strictly decreasing about some point $x_0 \in \mathbb{R}^N$. This result holds for the full range $1 < p < N$. The proof involves showing that the solutions are symmetric with respect to a hyperplane in any direction, eventually converging to a unique center of symmetry.

3. Uniqueness and Complete Classification:
   The authors demonstrate that $u \equiv v$. Consequently, the system reduces to the single critical $p$-Laplacian equation $-\Delta_p u = u^{p^*-1}$. By leveraging existing classification results for the single equation (e.g., from Sciunzi [59] and Vetois [71]), the paper provides the complete classification of positive solutions:
   $$u(x) = v(x) = \left[ \lambda^{\frac{1}{p-1}} \left( N^{\frac{1}{p}} \left( \frac{N-p}{p-1} \right)^{\frac{p-1}{p}} \right) \lambda^{\frac{p}{p-1}} + |x-x_0|^{\frac{p}{p-1}} \right]^{-\frac{N-p}{p}},$$
   where $\lambda > 0$ and $x_0 \in \mathbb{R}^N$ are arbitrary parameters.

Significance and Scope
The paper claims to extend the uniqueness results known for the semi-linear case ($p=2$) to the general quasi-linear case ($1 < p < N$). It specifically addresses the technical void left by the absence of conformal invariance and Kelvin transforms in the $p$-Laplacian setting. By establishing sharp asymptotic behaviors and handling the singular/degenerate nature of the operator separately, the work provides a rigorous foundation for the classification of solutions to critical quasi-linear systems. The results unify the understanding of such systems across the entire range of $p$, confirming that despite the increased complexity of the operator, the qualitative behavior (radial symmetry, monotonicity, and uniqueness up to scaling and translation) remains consistent with the semi-linear case.