Construction of solutions for some localized nonlinear Schrödinger equations

Abstract

For an N-body system of linear Schrödinger equation with space dependent interaction between particles, one would expect that the corresponding one body equation, arising as a mean field approximation, would have a space dependent nonlinearity. With such motivation we consider the following model of a nonlinear reduced Schrödinger equation with space dependent nonlinearity −ϕ + V (x)h (|ϕ| 2)ϕ = λϕ, where V (x) = −χ [−1,1] (x) is minus the characteristic function of the interval [−1, 1] and where h is any continuous strictly increasing function. In this article, for any negative value of λ we present the construction and analysis of the infinitely many solutions of this equation, which are localized in space and hence correspond to bound-states of the associated time-dependent version of the equation.

Ground state solutions for the fractional problems with dipole-type potential and critical exponent

“…In this study, we restrict our attention toH(i∇, x) with m = 0. As we see, equation (P S ) is related to the fractional Schrödinger equation: When s = 1, it is the well-known Schrödinger equation which arises in quite a few physical contexts such as quantum mechanics, nonlinear optics, quantum field theory and Hartree-Fock theory [1,15,16]. When s ∈ (0, 1) \ { 1 2 }, it is the fractional Schrödinger equation, which was introduced by Laskin (see [9]) owing to the extension of the Feynman path integral, from the Brownian-like to Lévy-like quantum mechanical paths.…”