A note on the inhomogeneous fractional nonlinear Schrödinger equation

Abstract

This paper investigates some well-posedness issues of the fractional inhomogeneous Schrödinger equation iu˙-(-Δ)γu=±|x|ρ|u|p-1u,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} i\\dot{u}-(-\\Delta )^\\gamma u=\\pm |x|^\\rho |u|^{p-1}u, \\end{aligned}$$\\end{document}where 0<γ<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\gamma <1$$\\end{document} and ρ<0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho <0$$\\end{document}. Here, one considers the inter-critical regime 0<sc:=N2-2γ+ρp-1<γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<s_c:=\\frac{N}{2}-\\frac{2\\gamma +\\rho }{p-1}<\\gamma $$\\end{document}, where sc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_c$$\\end{document} is the energy critical exponent, which is the only one real number satisfying ‖κ2γ+ρp-1u0(κ·)‖H˙sc=‖u0‖H˙sc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert \\kappa ^\\frac{2\\gamma +\\rho }{p-1}u_0(\\kappa \\cdot )\\Vert _{\\dot{H}^{s_c}}=\\Vert u_0\\Vert _{\\dot{H}^{s_c}}$$\\end{document}. In order to avoid a loss of regularity in Strichartz estimates, one assumes that the datum is spherically symmetric. First, using a sharp Gagliardo–Nirenberg-type estimate, one develops a local theory in the space H˙γ∩H˙sc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dot{H}^\\gamma \\cap \\dot{H}^{s_c}$$\\end{document}. Then, one investigates the LN(p-1)ρ+2γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{\\frac{N(p-1)}{\\rho +2\\gamma }}$$\\end{document} concentration of finite-time blow-up solutions bounded in H˙sc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dot{H}^{s_c}$$\\end{document}. Finally, one proves the existence of non-global solutions with negative energy. Since one considers the homogeneous Sobolev space H˙sc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dot{H}^{s_c}$$\\end{document}, the main difficulty here is to avoid the mass conservation law.