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{align*} \left{ \begin{aligned} &-\Delta u+\lambda u=(I_{\alpha}\ast F(u))f(u), \quad \quad \hbox{in }\mathbb{R}^{2},\ &\int_{\mathbb{R}^{2}}|u|^{2}dx=a^{2}, \end{aligned} \right. \end{align*} where $a>0$ is prescribed, $\lambda\in \mathbb{R}$, $\alpha\in(0,2)$, $I_{\alpha}$ denotes the Riesz potential, $\ast$ indicates the convolution operator, the function $f(t)$ has exponential growth in $\mathbb{R}^{2}$ and $F(t)=\int^{t}_{0}f(\tau)d\tau$. Using the Pohozaev manifold and variational methods, we establish the existence of normalized solutions to the above problem.