Dynamics of the non-radial energy-critical inhomogeneous NLS

We consider the focusing inhomogeneous nonlinear Schr\"odinger equation \ iₜ u + u + |x|^-b|u|^ u = 0onN, \ with =4-2bN-2, N=\3, 4, 5\ and 0<b \6-N{2, 4N\}. This paper establishes global well-posedness and scattering for the non-radial energy-critical case in Ḣ¹ (N). It extends the previous research by Murphy and the first author GM, which focused on the case (N, , b) = (3, 2, 1). The novelty here, beyond considering higher dimensions, lies in our assumption of the condition ₓ ₈\| u (t) \|₋ℂ<\| Q\|₋ℂ, which is weaker than the condition stated in Guzman. Consequently, if a solution has energy and kinetic energy less than the ground state Q at some point, then the solution is global and scatters. Moreover, we show scattering for the defocusing case. On the other hand, in this work, we also investigate the blow-up issue with nonradial data for N 3 in H¹ (RN). This implies that our result holds without classical assumptions such as spherically symmetric data or |x|u₀ L² (RN). \ Mathematics Subject Classification. 35A01, 35QA55, 35P25.