A logarithmic Choquard equation with critical exponential growth in ℝ ²

In this paper, we are paid attention to the following logarithmic Choquard equation −Δu+u=[I2∗F(u)]f(u),x∈R2,where I2=12πln⁡1|x| is the Newtonian kernel in dimension 2, F(u)=∫0uf(t)dt and f∈C1(R,R) is critical exponential growth with respect to Trudinger–Moser. Based on a variational framework proposed in Liu et al. (J Differ Equ 2022;328), we obtain a positive finite energy solution of the above logarithmic Choquard equation via the variational methods. Furthermore, we use another approach to investigate the planar logarithmic Choquard equations involving critical exponential growth.