Normalized solutions for a Choquard equation with exponential growth in $\mathbb {R}^{2}$

Abstract

In this paper, we study the existence of normalized solutions to the following nonlinear Choquard equation with exponential growth

$$\begin{aligned} \left\{ \begin{aligned}&-\Delta u+\lambda u=(I_{\alpha }*F(u))f(u), \quad \quad \hbox {in }\mathbb {R}^{2},\\&\mathop \int \limits _{\mathbb {R}^{2}}|u|^{2}\textrm{d}x=a^{2}, \end{aligned} \right. \end{aligned}$$

where $a>0$ is prescribed, $\lambda \in \mathbb {R}$, $\alpha \in (0,2)$, $I_{\alpha }$ denotes the Riesz potential, $*$ indicates the convolution operator, the function f(t) has exponential growth in $\mathbb {R}^{2}$ and $F(t)=\mathop \int \limits ^{t}_{0}f(\tau )\textrm{d}\tau$. Using the Pohozaev manifold and variational methods, we establish the existence of normalized solutions to the above problem.