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

TL;DR

This paper proves the existence of normalized solutions for a nonlinear Choquard equation with exponential growth in two-dimensional space using variational methods and the Pohozaev manifold.

Contribution

It introduces a new approach to find normalized solutions for Choquard equations with exponential nonlinearity in b2, expanding the understanding of such equations.

Findings

Existence of solutions established using variational methods.

Application of the Pohozaev manifold to normalized solutions.

Solutions found for prescribed L^2 norm in b2.

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.