Normalized solutions to nonlinear Schrodinger equations with competing Hartree-type nonlinearities

Abstract

In this paper, we consider solutions to the following nonlinear Schrodinger equation with competing Hartree-type nonlinearities, -Delta u + lambda u = (|x|(-gamma)(1) * |u|(2))u - (|x|(-gamma)(2) * |u|(2))u in R-N, under the L-2-norm constraint integral(N)(R) |u|(2) dx = c > 0, where N >= 1, 0 < gamma(2) < gamma(1) < min{N,4}, and lambda is an element of R appearing as Lagrange multiplier is unknown. First, we establish the existence of ground states in the mass subcritical, critical, and supercritical cases. Then, we consider the well-posedness and dynamical behaviors of solutions to the Cauchy problem for the associated time-dependent equations.