A note on inhomogeneous fractional Schrödinger equations

Abstract

AbstractWe study some energy well-posedness issues of the Schrödinger equation with an inhomogeneous mixed nonlinearity and radial data$$ i\dot{u}-(-\Delta )^{s} u \pm \vert x \vert ^{\rho} \vert u \vert ^{p-1}u\pm \vert u \vert ^{q-1}u=0, \quad 0< s< 1, \rho \neq 0, p,q>1. $$iu˙−(−Δ)su±|x|ρ|u|p−1u±|u|q−1u=0,0<s<1,ρ≠0,p,q>1.Our aim is to treat the competition between the homogeneous term$|u|^{q-1}u$|u|q−1uand the inhomogeneous one$|x|^{\rho}|u|^{p-1}u$|x|ρ|u|p−1u. We simultaneously treat two different regimes,$\rho >0$ρ>0and$\rho <-2s$ρ<−2s. We deal with three technical challenges at the same time: the absence of a scaling invariance, the presence of the singular decaying term$|\cdot |^{\rho}$|⋅|ρ, and the nonlocality of the fractional differential operator$(-\Delta )^{s}$(−Δ)s. We give some sufficient conditions on the datum and the parametersN,s,ρ,p,qto have the global versus nonglobal existence of energy solutions. We use the associated ground states and some sharp Gagliardo–Nirenberg inequalities. Moreover, we investigate the$L^{2}$L2concentration of the mass-critical blowing-up solutions. Finally, in the attractive regime, we prove the scattering of energy global solutions. Since there is a loss of regularity in Strichartz estimates for the fractional Schrödinger problem with nonradial data, in this work, we assume that$u_{|t=0}$u|t=0is spherically symmetric. The blowup results use ideas of the pioneering work by Boulenger el al. (J. Funct. Anal. 271:2569–2603, 2016).