  In this paper we study the equation \[ -\Delta u +(\log
|\cdot|*|u|^2)u=(\log|\cdot|*|u|^q)|u|^{q-2}u, \qquad \hbox{ in }\mathbb{R}^2,
\] where $8/3 < q < 4$. By means of variational arguments, we find infinitely
many radially symmetric classical solutions. The main difficulties rely on the
competition between the two nonlocal terms and on the presence of logarithmic
kernels, which have not a prescribed sign. In addition, in order to find finite
energy solutions, a suitable functional setting analysis is required.
