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Abstract The paper deals with the existence of normalized solutions for the following Schrödinger–Poisson system with L² L 2 -constraint: aligned \ array{ll - u+ u+ (| |*u²) u= (e^u²-1-u²) u, & x {R}², \\ {ₑ²}u²dx=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 >0 μ > 0, {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.
Chen et al. (Fri,) studied this question.