A note on inhomogeneous fractional Schrödinger equations

Abstract

We study some energy well-posedness issues of the Schrödinger equation with an inhomogeneous mixed nonlinearity and radial data iu˙−(−Δ)su±|x|ρ|u|p−1u±|u|q−1u=0,0<s<1,ρ≠0,p,q>1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ i\\dot{u}-(-\\Delta )^{s} u \\pm \\vert x \\vert ^{\\rho} \\vert u \\vert ^{p-1}u\\pm \\vert u \\vert ^{q-1}u=0, \\quad 0< s< 1, \\rho \ eq 0, p,q>1. $$\\end{document} Our aim is to treat the competition between the homogeneous term |u|^{q-1}u and the inhomogeneous one |x|^{rho}|u|^{p-1}u. We simultaneously treat two different regimes, rho >0 and rho <-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}, and the nonlocality of the fractional differential operator (-Delta )^{s}. We give some sufficient conditions on the datum and the parameters N, s, ρ, p, q to 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} concentration 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} is spherically symmetric. The blowup results use ideas of the pioneering work by Boulenger el al. (J. Funct. Anal. 271:2569–2603, 2016).