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, $$\mid {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha \mathrm{/}{q}^{\mathrm{\prime }}-n}\mathrm{\mid }y{\mathrm{\mid }}^{\alpha \mathrm{/}{q}^{\mathrm{\prime }}}g(x)h(y)dxdy\mid \ge C_{n,\alpha ,p,q^{\mathrm{\prime }}}\mathrm{\parallel }g{\mathrm{\parallel }}_{L^{q^{\mathrm{\prime }}}({\mathbb{R}}^n)}\mathrm{\parallel }h{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}$$ holds for any nonnegative functions $g\in L^{q^{\mathrm{\prime }}}({\mathbb{R}}^n)$ and $h\in L^p({\mathbb{R}}^n)$, where $n\ge 1$, $p,q^{\mathrm{\prime }}\in (0,1)$, $\alpha >n$ satisfying $1\mathrm{/}p+1\mathrm{/}{q}^{\mathrm{\prime }}-2\alpha \mathrm{/}(q^{\mathrm{\prime }}n)=1$. 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 $$\{\begin{array}{c}u(x)={\int }_{{\mathbb{R}}^n}\mathrm{\mid }x-y{\mathrm{\mid }}^{\beta -n}{v}^{-p_2}(y)\mathrm{\mid }y{\mathrm{\mid }}^{\beta }dy,v(x)={\int }_{{\mathbb{R}}^n}\mathrm{\mid }x-y{\mathrm{\mid }}^{\beta -n}{u}^{-p_1}(y)\mathrm{\mid }x{\mathrm{\mid }}^{\beta }dy\mathrm{.}\end{array}$$ We obtain necessary conditions for the existence of positive solutions, and investigate their integrability and asymptotic behavior when $\mathrm{\mid }x\mathrm{\mid }\to 0$ and $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$.