Dynamics of the non-radial energy-critical inhomogeneous NLS

We consider the focusing inhomogeneous nonlinear Schr\"odinger equation \[ i∂_t u + Δ u + |x|⁻ᵇ|u|^α u = 0{on}×^N, \] with α=4-2b/N-2, N=\3,4,5\ and 0<b≤ min\6-N/2,4/N\}. 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,α,b)=(3,2,1). The novelty here, beyond considering higher dimensions, lies in our assumption of the condition t∈ I\|∇ u(t)\|L²<\|∇ Q\|L², 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.