Existence of normalized solutions for the planar Schrödinger-Poisson system with exponential critical nonlinearity

Abstract

In the present work, we are concerned with the existence of normalized solutions to the following Schrödinger-Poisson System $$ \left\{ \begin{array}{ll} -\Delta u + \lambda u + \mu (\ln|\cdot|\ast |u|^{2})u = f(u) \ \text{ in } \mathbb{R}^2 , \\ \int_{\mathbb R^2} |u(x)|^2 dx = c,\ c > 0 , \end{array} \right. $$ for $\mu \in \mathbb{R} $ and a nonlinearity $f$ with exponential critical growth. Here, $\lambda\in \mathbb{R}$ stands as a Lagrange multiplier and it is part of the unknown. Our main results extend and/or complement some results found in [3] and [13].