A note on the inhomogeneous fractional nonlinear Schrödinger equation

Abstract

AbstractThis paper investigates some well-posedness issues of the fractional inhomogeneous Schrödinger equation $$\begin{aligned} i\dot{u}-(-\Delta )^\gamma u=\pm |x|^\rho |u|^{p-1}u, \end{aligned}$$ i u ˙ - ( - Δ ) γ u = ± | x | ρ | u | p - 1 u , where $$0<\gamma <1$$ 0 < γ < 1 and $$\rho <0$$ ρ < 0 . Here, one considers the inter-critical regime $$0<s_c:=\frac{N}{2}-\frac{2\gamma +\rho }{p-1}<\gamma $$ 0 < s c : = N 2 - 2 γ + ρ p - 1 < γ , where $$s_c$$ s c is the energy critical exponent, which is the only one real number satisfying $$\Vert \kappa ^\frac{2\gamma +\rho }{p-1}u_0(\kappa \cdot )\Vert _{\dot{H}^{s_c}}=\Vert u_0\Vert _{\dot{H}^{s_c}}$$ ‖ κ 2 γ + ρ p - 1 u 0 ( κ · ) ‖ H ˙ s c = ‖ u 0 ‖ 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 $$\dot{H}^\gamma \cap \dot{H}^{s_c}$$ H ˙ γ ∩ H ˙ s c . Then, one investigates the $$L^{\frac{N(p-1)}{\rho +2\gamma }}$$ L N ( p - 1 ) ρ + 2 γ concentration of finite-time blow-up solutions bounded in $$\dot{H}^{s_c}$$ H ˙ s c . Finally, one proves the existence of non-global solutions with negative energy. Since one considers the homogeneous Sobolev space $$\dot{H}^{s_c}$$ H ˙ s c , the main difficulty here is to avoid the mass conservation law.