STATIONARY WAVES WITH PRESCRIBED L-2-NORM FOR THE PLANAR SCHRODINGER-POISSON SYSTEM

The paper deals with the existence of standing wave solutions for the Schrodinger- Poisson system with prescribed mass in dimension N = 2. This leads to investigating the existence of normalized solutions for an integrodifferential equation involving a logarithmic convolution potential, namely, + Delta u + lambda u + gamma (log vertical bar. vertical bar * vertical bar u vertical bar(2)) u = a vertical bar u vertical bar(p-2)u in R-2, integral(R2) vertical bar u vertical bar(2)dx = c, where c > 0 is a given real number. Under different assumptions on gamma is an element of R, a is an element of , p > 2, we prove several existence and multiplicity results. Here lambda is an element of 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 the normalized solutions.