Some remarks on the inhomogeneous biharmonic NLS equation

Abstract

We consider the inhomogeneous biharmonic nonlinear Schrödinger equation $$i{u}_t+{\Delta }^{2}u+\lambda |x{|}^{-b}|u{|}^{\alpha }u=0,$$ where $\lambda =\pm 1$ and $\alpha$, $b>0$. In the subctritical case, we improve the global well-posedness result obtained in \cite{GUZPAS} for dimensions $N=5,6,7$ in the Sobolev space $H^2({\mathbb{R}}^N)$. The fundamental tools to establish our results are the standard Strichartz estimates related to the linear problem and the Hardy-Littlewood inequality. Results concerning the energy-critical case, that is, $\alpha =\frac{8-2b}{N-4}$ are also reported. More precisely, we show well-posedness and a stability result with initial data in the critical space ${\dot{H}}^2$.

16 pages