Integrability and Symmetry of Positive Integrable Solutions for Weighted Wolff-Type Integral Systems

期刊论文
发表日期：

2024年4月8日

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Yan Bai,
Zexin Zhang,
Zhitao Zhang

摘要

In this paper, we are concerned with the following weighted integral system involving Wolff potential: 1.1 $\begin{aligned} \left\{ \begin{array}{lll} u(x)=R_1(x)W_{\beta ,\gamma }\left( \frac{v^q}{|y|^\sigma }\right) (x),\quad &{} u(x)>0,\quad &{} x\in \mathbb {R}^N,\\ v(x)=R_2(x)W_{\beta ,\gamma }\left( \frac{u^p}{|y|^\sigma }\right) (x),\quad &{} v(x)>0,\quad &{} x\in \mathbb {R}^N, \end{array}\right. \end{aligned}$ $\begin{matrix} \{\begin{matrix} u (x) = R_{1} (x) W_{β, γ} (\frac{v^{q}}{{| y |}^{σ}}) (x), & u (x) > 0, & x \in R^{N}, \\ v (x) = R_{2} (x) W_{β, γ} (\frac{u^{p}}{{| y |}^{σ}}) (x), & v (x) > 0, & x \in R^{N}, \end{matrix}) \end{matrix}$ where $\gamma >2$ $γ > 2$ , $\beta >0$ $β > 0$ , $0<\sigma<\beta \gamma <N$ $0 < σ < β γ < N$ , $p,q>\gamma -1$ $p, q > γ - 1$ , with $\frac{\gamma -1}{p+\gamma -1}$ $\frac{γ - 1}{p + γ - 1}$ , $\frac{\gamma -1}{q+\gamma -1}<\frac{N-\beta \gamma }{N}$ $\frac{γ - 1}{q + γ - 1} < \frac{N - β γ}{N}$ , $\frac{\gamma -1}{p+\gamma -1}+\frac{\gamma -1}{q+\gamma -1}=\frac{N-\beta \gamma +\sigma }{N}$ $\frac{γ - 1}{p + γ - 1} + \frac{γ - 1}{q + γ - 1} = \frac{N - β γ + σ}{N}$ , $R_1,R_2$ $R_{1}, R_{2}$ are double bounded in $\mathbb {R}^N$ $R^{N}$ and $\begin{aligned} W_{\beta ,\gamma }(h)(x):=\int _{0}^{\infty }\left[ \frac{\int _{B_t(x)}h(y)dy}{t^{N-\beta \gamma }}\right] ^{\frac{1}{\gamma -1}}\frac{dt}{t}. \end{aligned}$ $\begin{matrix} W_{β, γ} (h) (x) : = \int_{0}^{\infty} {[\frac{\int_{B_{t} (x)} h (y) d y}{t^{N - β γ}}]}^{\frac{1}{γ - 1}} \frac{dt}{t} . \end{matrix}$ Firstly, by applying Minkowski’s inequality and the regularity lifting lemma, we prove the optimal integrability, boundedness and vanishing property at infinity of integrable solutions for the system. Secondly, we use the method of moving planes in integral forms to prove the radial symmetry of the integrable solutions when $R_1\equiv R_2\equiv 1$ $R_{1} \equiv R_{2} \equiv 1$ in $\mathbb {R}^N$ $R^{N}$ .