Linear Profile Decomposition for the Nonlinear Schrödinger Equation in Exterior Domains

This paper establishes a linear profile decomposition for bounded sequences in the homogeneous Dirichlet–Sobolev space H ̇_D^1 (Ω), where Ω=R3/O is the exterior of a smooth, compact obstacle O⊂R3. Given a bounded sequence {fn}⊂H31 (Ω), we prove that, after passing to a subsequence, it decomposes as fn=∑_(j=1)^Jϕ_n^j +w_n^J, where the profiles {ϕ_n^j} are asymptotically orthogonal and the remainder w_n^J vanishes in all Strichartz spaces L_t^q L_x^r as J→∞. The decomposition satisfies an exact energy identity ∥∇f_n ∥_(L^2)^2=∑_j^∥ ∇ϕ_n^j ∥_(L^2)^2+∥∇w_n^J ∥_(L^2)^2+o(1). Four distinct geometric concentration regimes are identified according to the behaviour of the scale sequence {λ_n^j} relative to the distance d(x_n^j ) to ∂Ω: profiles localised inside Ω; profiles dispersing to spatial infinity (limiting domain R3); profiles concentrating deep inside Ω away from the boundary; and profiles concentrating near ∂Ω (limiting domain: a half-space). As an application, we prove small-data scattering in H ̇_D^1 (Ω) for the defocusing, energy-subcritical NLS i∂_t u+Δ_Ω u=|u|^(p-1) u with 1