In this paper we study the following nonlinear Choquard equation $-\Delta u+u=\left(ln\frac{1}{\left|x\right|}\ast F\left(u\right)\right)f\left(u\right),\text{in}{\mathbb{R}}^2,$where $f\in C^1(\mathbb{R},\mathbb{R})$ 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 $H^1\left({\mathbb{R}}^2\right)$. We give a new proof and at the same time extend part of the results established in (Cassani-Tarsi, Calc.Var.PDE, 2021) [11].

中文翻译：

---

具有指数非线性的平面对数 Choquard 方程的正解

在本文中，我们研究了以下非线性 Choquard 方程$-\Delta 你+你=\left(因\frac{1}{\left|X\right|}*F（你）\right)F（你）,\text{在}{\mathbb{右}}^2,$在哪里$F\epsilon C^1（\mathbb{右},\mathbb{右}）$和$F$是非线性的原语$F$消失于零。我们使用渐近逼近方法来建立标准 Sobolev 空间中上述问题正解的存在性$H^1（{\mathbb{右}}^2）$。我们给出了新的证明，同时扩展了(Cassani-Tarsi, Calc.Var.PDE, 2021) [11]中建立的部分结果。