A minimization problem for the Nonlinear Schr¨odinger-Poisson type Equation

Abstract

In this paper we consider the stationary solutions of the Schr¨odinger-Poisson equation:

it + − (|x|−1 | |2) + | |p−2 = 0 in R3. We are interested in the existence of standing waves, that is solutions of type (x, t) = u(x)e−i!t, where ! 2 R, with fixed L2 −norm. Then we are reduced to a constrained minimization problem. The main difficulty is the compactness of the minimizing sequences since the related functional is invariant  y translations. By using some abstract results, we give a positive answer, showing that the minimum of the functional is achieved on small L2 −spheres in the case 2 < p < 3 and large L2 − spheres in the case 3 < p < 10/3. The results exposed here can be found with more details in [6] and [7].