On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity

Orbital stability of standing waves of a class of fractional Schrödinger equations with Hartree-type nonlinearity

“…Let us denote the nonlinear term [V (|x|) ⋆ G(φ)]G ′ (φ) by N (φ). Since the well-posedness of the case µ = 2, 2s = N − β was treated in [5], in this paper we consider the initial value problem S with µ ∈ 2, 1 + 2s+β…”

“…Before going further, let us recall the following weighted Strichartz estimate (see for instance Lemma 6.2 of [5] and Lemma 2 of [6]).…”

“…As noticed in the introduction of this paper, this problem has been studied in various situation depending on the value of s and the conditions on β and the integrand G in Ref. [5,12,18]. In order to prove the existence of critical points to the functional E, we start with the following claim Proposition 3.1.…”

“…The mathematical analysis of the fractional nonlinear Schrödinger equation has been growing continually during the last few decades. Many results have been obtained and we refer for instance to [5] and references therein. This paper deals with the analysis of the following Cauchy problem S :…”

“…The particular case µ = 2 and 2s = N − β was treated in Ref. [5] and for lightness of the proofs, we shall sometimes omit it and focus on the case µ ∈ 2, 1 + 2s+β N . The proof of the existence part of Theorem 1.1 is based on a classical contraction argument and the conservation laws associated to the dynamics of the system S .…”