Positive solutions to the planar logarithmic Choquard equation via asymptotic approximation

Abstract

In this paper we study the following nonlinear Choquard equation $$ -\Delta u+u=\left(\ln\frac{1}{|x|}\ast F(u)\right)f(u),\quad\text{ in }\,\mathbb{R}^2, $$ where $f\in C^1(\mathbb{R})$ and $F$ is the primitive of the nonlinearity $f$ vanishing at zero. We use an asymptotic approximation approach to establish the existence of positive solutions to the above problem in the standard Sobolev space $H^1(\mathbb{R}^2)$. We give a new proof and at the same time extend part of the results established in [Cassani-Tarsi, Calc. Var. P.D.E. (2021)].