Multiple normalized solutions for the planar Schrödinger–Poisson system with critical exponential growth

The paper deals with the existence of normalized solutions for the following Schrödinger–Poisson system with L² L 2 -constraint: $${aligned} \{ {array}{ll} -Δ u+λ u+μ ( log |· |*u^2) u=( eu^2-1-u^2) u, &{} x∈ {R}^2, \\ ∫ _{{R}^2}u^2dx=c, \\ {array} . {aligned}$$ - Δ u + λ u + μ log | · | ∗ u 2 u = e u 2 - 1 - u 2 u , x ∈ R 2 , ∫ R 2 u 2 d x = c , where $$μ >0$$ μ > 0 , $$λ ∈ {R}$$ λ ∈ 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 L² 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⁻²u a | u | p - 2 u in the case of $$a>0$$ a > 0 and $$p>4$$ 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 L² L 2 -constrained problems with critical exponential growth, and the new underlying ideas are of future development and applicability.