Existence of two normalized solutions for a Choquard equation with exponential growth and an \(L^2\)-subcritical perturbation

This paper is concerned with the existence of normalized solutions for the following class of Choquard elliptic problems: \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u + \lambda u = (I_{\alpha } *F(u))f(u) + \mu (I_{\alpha } *|u|^{q})|u|^{q-2}u, \text { in } \mathbb {R}^{2}, \\ \displaystyle \int _{\mathbb {R}^2}|u|^2 = a, \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}-\mathrm{\Delta }u+\lambda u=(I_{\alpha }\ast F(u))f(u)+\mu (I_{\alpha }{\ast |u|}^q{)|u|}^{q-2}u,\text{in}{\mathbb{R}}^2,\hfill \\ {\displaystyle {\int }_{{\mathbb{R}}^2}{|u|}^2=a,}\hfill \end{array}\right)\end{array}$$ where \(a>0\) $a>0$ , \(I_{\alpha }\) $I_{\alpha }$ is the Riesz potential, \(*\) $\ast$ represents the convolution operator, f has exponential critical growth in \(\mathbb {R}^2\) ${\mathbb{R}}^2$ , \(F(t)= \int _{0}^t f(s)ds\) $F(t)={\int }_0^{t}f(s)ds$ , and \(1 + \frac{\alpha }{2}< q < 2 + \frac{\alpha }{2}\) $1+\frac{\alpha }{2}<q<2+\frac{\alpha }{2}$ . By variational methods, we prove the existence of two normalized solutions: one with negative energy and one with positive energy.