Concentrating normalized solutions to planar Schrödinger–Poisson system with steep potential well: critical exponential case

In this paper, we study the following class of planar Schrödinger–Poisson problems \(\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\Delta u +\lambda V(x)u +\gamma (\ln |\cdot |*|u|^{2})u =\mu u+\kappa f(u)~\text {in}~ \mathbb {R}^2, \\ \displaystyle u(x)>0 ~\text {in}~ {\mathbb {R}}^2, \\ \displaystyle \int _{\mathbb {R}^2} |u(x)|^2 dx = a^2, \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle -\mathrm{\Delta }u+\lambda V(x)u+\gamma (ln|·{|\ast |u|}^2)u=\mu u+\kappa f(u)\text{in}{\mathbb{R}}^2,}\hfill \\ {\displaystyle u(x)>0\text{in}{\mathbb{R}}^2,}\hfill \\ {\displaystyle {\int }_{{\mathbb{R}}^2}{|u(x)|}^{2}dx=a^2,}\hfill \end{array}\right)\end{array}$$ where \(a > 0\) $a>0$ , \(\mu \in {\mathbb {R}}\) $\mu \in \mathbb{R}$ is an unknown parameter appearing as a Lagrange multiplier, \(\lambda ,\gamma ,\kappa >0\) $\lambda ,\gamma ,\kappa >0$ are parameters, \(V\in \mathcal {C}({\mathbb {R}}^2,{\mathbb {R}}^+)\) $V\in \mathcal{C}({\mathbb{R}}^2,{\mathbb{R}}^{+})$ admits a potential well \(\Omega \triangleq \text {int}\, V^{-1}(0)\) $\mathrm{\Omega }\triangleq \text{int}{V}^{-1}(0)$ and f is a continuous function having critical exponential growth at infinity in the Trudinger-Moser sense. Owing to some technical tricks adopted in Alves and Shen (On existence of positive solutions to some classes of elliptic problems in the hyperbolic space, Submitted for publication), Shen and Squassina (Existence and concentration of normalized solutions for p-Laplacian equations with logarithmic nonlinearity, http://arxiv.org/abs/2403.09366), we are able to obtain the existence and concentrating behavior of positive normalized solutions for sufficiently large \(\lambda \) $\lambda$ using variational method.