Normalized solutions to an exponential growth Choquard equation driven by mixed local-nonlocal operator in $\mathbb{R}^2$

Abstract

In this article, we study the existence of normalized solutions to the following mixed nonlinear Choquard equation with exponential growth \begin{align*} \left\{ \begin{aligned} \mathcal{L}u+\lambda u \; &=\; \Lambda(I_{\alpha}\ast F(u))F'(u), \quad \text{in }\mathbb{R}^{2}, \int_{\mathbb{R}^{2}}|u|^{2}\,dx \; &=\; a^{2}, \end{aligned} \right. \end{align*} where $\mathcal{L}= -\Delta+(-\Delta)^s$, $0 0$, $I_{\alpha}$ is the Riesz potential of order $\alpha\in (0,2)$, $\Lambda>0$ is a parameter and $\lambda\in \mathbb{R}$ appears as a Lagrange multiplier. Here, the nonlinearity $F$ has exponential growth in $\mathbb{R}^{2}$. Using variational methods, we prove the existence of normalized solution in the Poho\v{z}aev manifold. Moreover, we discuss the regularity result and the construction of the Poho\v{z}aev identity, essential for the existence. \keywords{Normalized solutions; Nonlinear Schr\"odinger equations; Choquard nonlinearity; Critical exponential growth; Trudinger-Moser inequality}