Reversed inequality of the Herbst-type and the related Euler-Lagrange system

In 2008, Beckner (Proc. Amer. Math. Soc. 136(5), 1871-1885) proved two inequalities of the Herbst type, which are the critical forms of the Stein-Weiss inequality. In 2018, Chen et al. (Tran. Amer. Math. Soc. 370(12), 8429-8450) established the reversed Stein-Weiss inequality. In this paper, we are concerned about its critical case and give a reversed Herbst inequality. Namely, $$\left|\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}|x-y|^{\alpha/q'-n}|y|^{\alpha/q'}g(x)h(y)dxdy\right| \geq C_{n,\alpha,p,q'}\|g\|_{L^{q'}(\mathbb{R}^n)}\|h\|_{L^p(\mathbb{R}^n)}$$ holds for any nonnegative functions $g \in L^{q'}(\mathbb{R}^n)$ and $h \in L^p(\mathbb{R}^n)$, where $n\geq 1$, $p, q' \in (0,1)$, $\alpha>n$ satisfying ${1}/{p}+{1}/{q'}-{2\alpha}/(q'n)=1$. Such an inequality is not covered by the reversed Stein-Weiss inequality. Meanwhile, we prove the existence of extremal functions of this inequality. Finally, we study the Euler-Lagrange system satisfied by those extremal functions $$\left\{\begin{matrix} u(x)=\int_{\mathbb{R}^n}|x-y|^{\beta-n}v^{-p_2}(y)|y|^{\beta}dy, v(x)=\int_{\mathbb{R}^n}|x-y|^{\beta-n}u^{-p_1}(y)|x|^{\beta}dy. \end{matrix}\right.$$ We obtain necessary conditions for the existence of positive solutions, and investigate their integrability and asymptotic behavior when $|x| \to 0$ and $|x| \to \infty$.