In this paper, we are concerned with the following integral system $$\left\{\begin{array}{cc}{u}_i(x)={\int }_{{ℝ}^n}\frac{u_{i+1}^{p_{i+1}}(y)}{|x-y{|}^{n-\alpha }}dy,\hfill & i=1,2,\dots ,m-1,\hfill \\ u_m(x)={\int }_{{ℝ}^n}\frac{u_1^{p_1}(y)}{|x-y{|}^{n-\alpha }}dy,\hfill & m\ge 1,n\ge 1,\hfill \end{array}\right.$$ where $u_i>0$, $0<\alpha <n$, and $p_i>1$ ($i=1,2,\dots ,m$). When $m\in \{1,2\}$, such an integral system is associated with the best constants of the Hardy–Littlewood–Sobolev inequality. Chen, Li and their cooperators obtained optimal integrability intervals of the finite energy solutions by an argument of contraction and shrinking operators. This result is helpful to well understand the classification of the extremal functions of the Hardy–Littlewood–Sobolev inequality. The critical condition plays a key role in their work. In this paper, we study optimal integrability intervals when the positive solutions have some initial integrability. Now, the critical condition is not necessary, and we apply a weaker condition, the Serrin-type condition, to establish some important relations of exponents which come into play to lift the regularity. In addition, we also generalize this result to the case of $m\in {\mathbb{Z}}^{+}$.

在本文中，我们关注以下积分系统$$\left\{\begin{array}{cc}{你}_{\mathrm{一世}}(X)={\int }_{{ℝ}^n}\frac{{你}_{\mathrm{一世}+1}^{p_{\mathrm{一世}+1}}(\mathrm{是的})}{|X-\mathrm{是的}{|}^{n-\alpha }}d\mathrm{是的},\hfill & \mathrm{一世}=1,2,\dots ,米-1,\hfill \\ {你}_{米}(X)={\int }_{{ℝ}^n}\frac{{你}_1^{p_1}(\mathrm{是的})}{|X-\mathrm{是的}{|}^{n-\alpha }}d\mathrm{是的},\hfill & 米\ge 1,n\ge 1,\hfill \end{array}\right.$$在哪里${你}_{\mathrm{一世}}>0$,$0<\alpha <n$， 和$p_{\mathrm{一世}}>1$($\mathrm{一世}=1,2,\dots ,米$）。什么时候$米\in \{1,2\}$，这样的积分系统与 Hardy-Littlewood-Sobolev 不等式的最佳常数相关。Chen、Li 和他们的合作者通过收缩和收缩算子的论证获得了有限能量解的最优可积区间。该结果有助于更好地理解 Hardy-Littlewood-Sobolev 不等式的极值函数的分类。危急情况在他们的工作中起着关键作用。在本文中，我们研究了当正解具有一定初始可积性时的最优可积区间。现在，临界条件不是必需的，我们应用一个较弱的条件，即 Serrin 型条件，来建立一些重要的指数关系，这些关系可以用来提升正则性。此外，我们还将这个结果推广到$米\in {\mathbb{Z}}^{+}$.