The paper deals with the existence of normalized solutions for the following Schrödinger–Poisson system with \(L^2\) -constraint: \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\lambda u+\mu \left( \log |\cdot |*u^2\right) u=\left( e^{u^2}-1-u^2\right) u, &{} x\in {\mathbb {R}}^2, \\ \int _{{\mathbb {R}}^2}u^2\textrm{d}x=c, \\ \end{array} \right. \end{aligned}\) where \(\mu >0\) , \(\lambda \in {\mathbb {R}}\) will arise as a Lagrange multiplier and the nonlinearity enjoys critical exponential growth of Trudinger-Moser type. By specifying explicit conditions on the energy level c, we detect a geometry of local minimum and a minimax structure for the corresponding energy functional, and prove the existence of two solutions, one being a local minimizer and one of mountain-pass type. In particular, to catch a second solution of mountain-pass type, some sharp estimates of energy levels are proposed, suggesting a new threshold of compactness in the \(L^2\) -constraint. Our study extends and complements the results of Cingolani–Jeanjean (SIAM J Math Anal 51(4): 3533-3568, 2019) dealing with the power nonlinearity \(a|u|^{p-2}u\) in the case of \(a>0\) and \(p>4\) , which seems to be the first contribution in the context of normalized solutions. Our model presents some new difficulties due to the intricate interplay between a logarithmic convolution potential and a nonlinear term of critical exponential type and requires a novel analysis and the implementation of new ideas, especially in the compactness argument. We believe that our approach will open the door to the study of other \(L^2\) -constrained problems with critical exponential growth, and the new underlying ideas are of future development and applicability.
