Scattering for the non-radial inhomogenous biharmonic NLS equation

Abstract

We consider the focusing inhomogeneous biharmonic nonlinear Schrödinger equation in $$H^2(\mathbb {R}^N)$$ , $$\begin{aligned} iu_t + \Delta ^2 u - |x|^{-b}|u|^{\alpha }u=0, \end{aligned}$$ when $$b > 0$$ and $$N \ge 5$$ . We first obtain a small data global result in $$H^2$$ , which, in the five-dimensional case, improves a previous result from Pastor and the second author. In the sequel, we show the main result, scattering below the mass-energy threshold in the intercritical case, that is, $$\frac{8-2b}{N}< \alpha <\frac{8-2b}{N-4}$$ , without assuming radiality of the initial data. The proof combines the decay of the nonlinearity with Virial-Morawetz-type estimates to avoid the radial assumption, allowing for a much simpler proof than the Kenig-Merle roadmap.