Critical hyperbola for a mixed Lane-Emden system: existence and nonexistence

This paper is devoted to a mixed Lane-Emden system with distinct types of nonlinearities: $ \left\lbrace \begin{aligned} &-\Delta u = v^q \quad \text{in} \quad \mathbb{R}^N, \\ &-\Delta v = \bigg(\int_{\mathbb{R}^N}\frac{ u^{p}(y)}{|x-y|^{N-\mu}}dy\bigg)u^{p-1} \quad \text{in} \quad \mathbb{R}^N, \end{aligned} \right. $ where the dimension $ N\geq3 $, $ 00 $. To the best of our knowledge, this is the first work addressing a class of Lane-Emden type system that equipped with both local and nonlocal nonlinearities. We investigate the critical and subcritical cases with respect to the critical hyperbola$ \frac{N+\mu}{2Np}+\frac{1}{q+1} = \frac{N-2}{N}. $For the critical case, i.e. $ (p, q) $ lies on the critical hyperbola, by reducing the system to an equivalent scalar equation, we establish the existence of positive ground state solutions. We also obtain the symmetry, monotonicity, regularity, and asymptotic properties. For the subcritical case, i.e. $ (p, q) $ lies below the critical hyperbola, we establish some nonexistence results via the method of integral estimates.