Abstract
Abstract In this paper, we study blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation i ∂ t ψ − ( − Δ ) s ψ + ( I α ∗ | ψ | p ) | ψ | p − 2 ψ = 0. $$\begin{array}{} \displaystyle i\partial_t\psi- (-{\it\Delta})^s \psi+(I_\alpha \ast |\psi|^{p})|\psi|^{p-2}\psi=0. \end{array}$$ By using localized virial estimates, we firstly establish general blow-up criteria for non-radial solutions in both L 2-critical and L 2-supercritical cases. Then, we show existence of normalized standing waves by using the profile decomposition theory in Hs . Combining these results, we study the strong instability of normalized standing waves. Our obtained results greatly improve earlier results.
Highlights
Over the past decade, there has been a great deal of interest in studying the fractional Schrödinger equation (NLS)i∂t ψ = (−∆)s ψ + f (ψ), (1.1)where < s < and f (ψ) is the nonlinearity
In this paper, we study blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation i∂t ψ − (−∆)s ψ + (Iα * |ψ|p)|ψ|p− ψ =
We study the strong instability of normalized standing waves
Summary
There has been a great deal of interest in studying the fractional Schrödinger equation (NLS). The fractional di erential operator (−∆)s is de ned by (−∆)s ψ = F− [|ξ | sF(ψ)], where F and F− are the Fourier transform and inverse Fourier transform, respectively. The fractional NLS (1.1) was rst deduced by Laskin in [29, 30] by extending the Feynman path integral from the Brownian-like to the Lévy-like quantum mechanical paths. The fractional NLS arises in the description of Bonson stars as well as in water wave dynamics We consider blow-up criteria and instability of normalized standing waves for the fractional nonlinear Schrödinger-Choquard equation i∂t ψ − (−∆)s ψ + (Iα * |u|p)|ψ|p− ψ = , (t, x) ∈ [ , T*) × RN ,
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