A note on the inhomogeneous fractional nonlinear Schrödinger equation

Abstract

This paper investigates some well-posedness issues of the fractional inhomogeneous Schrödinger equation

$$\begin{aligned} i\dot{u}-(-\Delta )^\gamma u=\pm |x|^\rho |u|^{p-1}u, \end{aligned}$$

where \(0<\gamma <1\) and \(\rho <0\). Here, one considers the inter-critical regime \(0<s_c:=\frac{N}{2}-\frac{2\gamma +\rho }{p-1}<\gamma \), where \(s_c\) is the energy critical exponent, which is the only one real number satisfying \(\Vert \kappa ^\frac{2\gamma +\rho }{p-1}u_0(\kappa \cdot )\Vert _{\dot{H}^{s_c}}=\Vert u_0\Vert _{\dot{H}^{s_c}}\). 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 \(\dot{H}^\gamma \cap \dot{H}^{s_c}\). Then, one investigates the \(L^{\frac{N(p-1)}{\rho +2\gamma }}\) concentration of finite-time blow-up solutions bounded in \(\dot{H}^{s_c}\). Finally, one proves the existence of non-global solutions with negative energy. Since one considers the homogeneous Sobolev space \(\dot{H}^{s_c}\), the main difficulty here is to avoid the mass conservation law.