In this paper, we study constraint minimizers of the following L 2 −critical minimization problem:and N denotes the mass of the particles in the Schrödinger-Poisson-Slater system. We prove that e(N) admits minimizers for N < N * ∶= ||Q|| 2 2 and, however, no minimizers for N > N * , where Q(x) is the unique positive solution of △u − u + u 7 3 = 0 in R 3 . Some results on the existence and nonexistence of minimizers for e(N * ) are also established. Further, when e(N * ) does not admit minimizers, the limit behavior of minimizers as N ↗ N * is also analyzed rigorously. KEYWORDS constraint minimizers, limit behavior, Schrödinger-Poisson-Slater system 4 3 u. From the physical point of view, we are interested in looking for solutions of Equation 1 with a prescribed L 2 -norm. Specifically, for any given constant N > 0, we look for solutions u N ∈ H 1 (R 3 ) with ||u N || 2 2 = N. Motivated by other studies, 8,14-16 taking N ∈ R in Equation 1 as a suitable Lagrange multiplier, a solution u N ∈ H 1 (R 3 ) of (1) Math Meth Appl Sci. 2017;40 7705-7721.wileyonlinelibrary.com/journal/mma

We study a stochastic nonlinear Schrödinger system with multiplicative white noise in energy space H1. Based on deterministic and stochastic Strichartz estimates, we prove the local well-posedness of a mild solution. Then, we prove the global well-posedness in the mass subcritical case and the defocusing case. For the mass subcritical case, we also investigate the global existence when the L2 norm of the initial value is small enough. We also study the blow-up phenomenon and give a sharp criterion via a general virial identity.

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