Existence and instability of normalized standing waves for the fractional Schrödinger equations in the <i>L</i>2-supercritical case
Binhua Feng, Jiajia Ren, Qingxuan Wang
Abstract
In this paper, we study the existence and instability of normalized standing waves for the fractional Schrödinger equation i∂tψ = (−Δ)sψ − f(ψ), where 0 &lt; s &lt; 1, f(ψ) = |ψ|pψ with 4sN&lt;p&lt;4sN−2s or f(ψ) = (|x|−γ*|ψ|2)ψ with 2s &lt; γ &lt; min{N, 4s}. To do this, we consider normalized solutions of the associated stationary equation (−Δ)su + ωu − f(u) = 0. By constructing a suitable submanifold of a L2-sphere and considering an equivalent minimizing problem, we prove the existence of normalized solutions. In particular, based on this equivalent minimizing problem, we can easily obtain the sharp threshold of global existence and blow-up for the time-dependent equation. Moreover, we can show that all normalized ground state standing waves are strongly unstable by blow-up. Our results are a complementary to the results of Peng and Shi [J. Math. Phys. 59, 011508 (2018)] and Zhang and Zhu [J. Dyn. Differ. Equations 29, 1017–1030 (2017)], where the existence and stability of normalized standing waves have been studied in the L2-subcritical case.