A logarithmic Choquard equation with critical exponential growth in ℝ2

Abstract

ABSTRACT In this paper, we are paid attention to the following logarithmic Choquard equation − Δ u + u = [ I 2 ∗ F ( u ) ] f ( u ) , x ∈ R 2 , where I 2 = 1 2 π ln ⁡ 1 | x | is the Newtonian kernel in dimension 2, F ( u ) = ∫ 0 u f ( t ) d t and f ∈ C 1 ( 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.