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+λu \; &=\; Λ(I_α\ast F(u))F&#39;(u), \quad \text{in }\mathbb{R}^{2}, \int_{\mathbb{R}^{2}}|u|^{2}\,dx \; &=\; a^{2}, \end{aligned} \right. \end{align*} where $\mathcal{L}= -Δ+(-Δ)^s$, $0<s<1$, $a>0$, $I_α$ is the Riesz potential of order $α\in (0,2)$, $Λ>0$ is a parameter and $λ\in \mathbb{R}$ 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 Pohožaev manifold. Moreover, we discuss the regularity result and the construction of the Pohožaev identity, essential for the existence. \keywords{Normalized solutions; Nonlinear Schrödinger equations; Choquard nonlinearity; Critical exponential growth; Trudinger-Moser inequality}