Radial Symmetry and Asymptotic Estimates for Positive Solutions to a Singular Integral Equation

Abstract

In this paper, we are concerned with the nonlinear singular integral equation\[ u(x) = |x|^\sigma \int_{R^n} \frac{u^p(y) \, dy}{|x-y|^{n-\alpha}},\]where $\alpha \in (0, n)$, $\sigma \in (\max \{-\alpha, \frac{\alpha-n}{2}\}, 0]$. Such an integral equation appears in the study of sharp constants of the Hardy-Sobolev inequality and the Hardy-Littlewood-Sobolev inequality. It is often used to describe the shapes of the extremal functions. If $0 \lt p \leq \frac{n}{n-\alpha-\sigma}$, there is not any positive solution to this equation. Under the assumption of $p = \frac{n+\alpha+2\sigma}{n-\alpha}$, we obtain an integrability result for the integrable solution $u$ (i.e., $u \in L^{\frac{2n}{n-\alpha}}(\mathbb{R}^n)$) of the integral equation. Such an integrable solution is radially symmetric and decreasing about $x_0 \in \mathbb{R}^n$. Furthermore, $x_0$ is also the origin if $\sigma \neq 0$. In addition, this integrable solution is blowing up with the rate $-\sigma$ when $|x| \to 0$. Moreover, if $n + p \sigma \gt 0$, then $u$ decays fast with the rate $n - \alpha - \sigma$ when $|x| \to \infty$.