GROUND STATE TO A NEW CLASS OF KIRCHHOFF-TYPE EQUATION WITH DOUBLY CRITICAL HARTREE-TYPE NONLINEARITIES

In this paper, we consider the following class of Kirchhoff-type equation with doubly critical Hartree-type nonlinearities <p class="disp_formula">$ \left\{ \begin{align}&amp; -(a+b\int _{\mathbb{R}^N}|\nabla u|^2\text{d}x)\Delta u+V(x)u=(I_{\alpha}*|u|^p)|u|^{p-2}u+(I_{\alpha}*|u|^q)|u|^{q-2}u,\,\, \text{in}\,\,\mathbb{R}^N,\\&amp; u\,\in\,H^1(\mathbb{R}^N), \end{align} \right. $ where $ a&gt;0 $, $ b\geq0 $, $ N\geq 3 $, $ \alpha\,\in\,(N-2,N) $, $ p=\frac{N+\alpha }{N-2} $, $ q=\frac{N+\alpha }{N} $, $ V:\,\mathbb{R}^N \rightarrow \mathbb{R} $ is a potential function and $ I_{\alpha} $ is a Riesz potential of order $ \alpha\,\in\,(N-2,N) $. Under certain assumptions on potential function $ V(x) $, we prove that the equation has at least a ground state solution by variational methods.