Radial symmetry of positive solutions of an integral system associated with the reversed Stein-Weiss inequality

Technical Summary: Radial Symmetry of Positive Solutions of an Integral System Associated with the Reversed Stein-Weiss Inequality

Problem Statement
This paper investigates the radial symmetry and monotonicity of positive solutions to a specific system of integral equations arising as the Euler-Lagrange system for extremal functions of the reversed Stein-Weiss inequality. The system is given by:
$$\begin{cases} u(x) = \int_{\mathbb{R}^n} \frac{|x|^\alpha v^{-p_2}(y) |y|^\beta}{|x-y|^\Lambda} dy, \\ v(x) = \int_{\mathbb{R}^n} \frac{|x|^\beta u^{-p_1}(y) |y|^\alpha}{|x-y|^\Lambda} dy, \end{cases}$$
where $n \ge 1$, $\Lambda > 0$, $p_1, p_2 > 1$, and the weights satisfy specific constraints involving $\alpha, \beta$. The authors consider solutions $(u, v)$ that are positive almost everywhere in $\mathbb{R}^n$ and belong to $L^\infty_{loc}(\mathbb{R}^n) \times L^\infty_{loc}(\mathbb{R}^n)$.

The problem extends previous work on the Hardy-Littlewood-Sobolev (HLS) and Stein-Weiss inequalities. While Chen, Li, and Ou established radial symmetry for standard HLS and Stein-Weiss systems using the method of moving planes, and Dou, Zhu, and Liu addressed the reversed HLS inequality, the radial symmetry for the reversed Stein-Weiss system (which involves double weights and negative exponents in the integrand) remained an open question.

Methodology
The authors employ the method of moving planes in integral form, a technique originally developed by Chen, Li, and Ou and later refined by Dou, Guo, and Zhu for systems with negative exponents. The proof proceeds through the following technical steps:

1. Auxiliary Functions and Regularity: To handle the double weights ($|x|^\alpha, |y|^\beta$) and the singularity at the origin, the authors introduce auxiliary functions:
   $$t(x) = |x|^{-\alpha}u(x), \quad w(x) = |x|^{-\alpha/p_1}u(x), \quad s(x) = |x|^{-\beta}v(x).$$
   They first establish the asymptotic behavior of the solutions $(u, v)$ at infinity and near the origin (Lemma 2.1). Crucially, they prove that the auxiliary functions $t$ and $s$ are continuous in $\mathbb{R}^n$ and differentiable in $\mathbb{R}^n \setminus \{0\}$, and derive precise estimates for their gradients (Lemmas 2.2 and 2.3). These estimates are necessary to manage the singularities near the origin during the moving plane process.

2. Integral Representation: The authors derive integral representations for the differences $t(x) - t_\lambda(x)$ and $s(x) - s_\lambda(x)$, where $x_\lambda$ denotes the reflection of $x$ across the plane $T_\lambda = \{x_1 = \lambda\}$. These representations involve the kernel $G(x, y) = |x_\lambda - y|^\Lambda - |x - y|^\Lambda$ (Lemma 2.4).

3. The Moving Plane Argument:

   • Starting Point: For sufficiently negative $\lambda$, the authors show that $t(x) \ge t_\lambda(x)$ and $s(x) \ge s_\lambda(x)$ hold in the region $\Sigma_\lambda = \{x \in \mathbb{R}^n \mid x_1 \le \lambda\}$. This relies on the asymptotic behavior of the solutions at infinity and the specific growth rates of the auxiliary functions.
   • Moving the Plane: The plane $T_\lambda$ is moved to the right as long as the inequalities hold. The authors define $\lambda_0 = \sup \{ \mu < 0 \mid \text{inequalities hold for all } \lambda < \mu \}$.
   • Contradiction: They assume $\lambda_0 < 0$ and derive a contradiction. By analyzing the integral representations and utilizing the strict positivity of the kernel $G(x, y)$ and the monotonicity of the power functions involved, they show that if the inequalities hold up to $\lambda_0$, they must strictly hold in a neighborhood beyond $\lambda_0$, unless the solution is already symmetric. The proof carefully handles the singularity at the origin by splitting the integration domains and using the established gradient estimates.
   • Conclusion of Symmetry: The contradiction implies $\lambda_0 = 0$. Since the direction of the $x_1$-axis is arbitrary, the solutions $w$ and $s$ (and consequently $u$ and $v$) are radially symmetric about the origin.

Key Contributions and Results
The main result is Theorem 1.1, which states that under the conditions $n \ge 1$, $\Lambda > 0$, $\min\{p_1, p_2\} > 1$, and specific bounds on the weights $\alpha$ and $\beta$, any pair of positive solutions $(u, v)$ to the system (1.12) is radially symmetric and monotonically increasing about the origin.

The paper's specific technical contributions include:

• Extending the method of moving planes to the reversed Stein-Weiss system, which features double weights and negative exponents ($v^{-p_2}, u^{-p_1}$).
• Introducing and utilizing specific auxiliary functions ($t, w, s$) to decouple the weights from the integral kernels, allowing the application of the moving plane method despite the lack of a direct reversed Stein-Weiss inequality for the system.
• Providing rigorous estimates for the gradients of the auxiliary functions near the origin and at infinity to ensure the validity of the integral estimates required for the moving plane argument.

Significance and Claims
The paper claims to resolve the question of radial symmetry for the extremal functions of the reversed Stein-Weiss inequality, a problem motivated by the work of Chen, Liu, Lu, and Tao (2018) who established the inequality but did not classify the symmetry of its extremals.

The authors position their work as a natural progression in the study of conformal equations and integral systems. They note that while the Stein-Weiss inequality does not directly apply to their system in the same way it did for the standard Stein-Weiss system (as in Jin and Li, 2006), the scheme developed by Dou, Guo, and Zhu for the reversed HLS inequality can be successfully adapted. The paper emphasizes that the double-weighted nature of the system presents a difficulty that requires the introduction of the auxiliary functions to handle the singularity at the origin effectively.

The work does not propose new applications or experimental setups but contributes to the theoretical understanding of the structure of solutions to nonlinear integral equations arising in mathematical physics and the theory of best constants in functional inequalities.