Stationary waves with prescribed L2-norm for the planar Schrödinger-Poisson system

The paper deals with the existence of standing wave solutions for the Schr"odinger-- Poisson system with prescribed mass in dimension N = 2. This leads to investigating the existence of normalized solutions for an integro-differential equation involving a logarithmic convolution potential, namely, -Delta u + lambda u + gamma igl( log | cdot | ast | u| 2igr) u = a| u| p 2u in R2, int R2 | u| 2dx = c, where c > 0 is a given real number. Under different assumptions on gamma in R, a in R, p > 2, we prove several existence and multiplicity results. Here lambda in R appears as a Lagrange parameter and is part of the unknowns. With respect to the related higher-dimensional cases, the presence of the logarithmic kernel, which is unbounded from above and below, makes the structure of the solution set much richer, forcing the implementation of new ideas to catch normalized solutions.