| Abstrakt: |
In this paper, we consider the inhomogeneous nonlinear Schrödinger equation i ∂ t u + Δ u = K (x) | u | α u , u (0) = u 0 ∈ H 1 (R N) , N ≥ 3 , | K (x) | + | x | | ∇ K (x) | ≲ | x | - b , 0 < b < min (2 , N - 2) , 0 < α < (4 - 2 b) / (N - 2) . We obtain novel results of global existence for oscillating initial data and scattering theory in a weighted L 2 -space for a new range α 0 (b) < α < (4 - 2 b) / N . The value α 0 (b) is the positive root of N α 2 + (N - 2 + 2 b) α - 4 + 2 b = 0 , which extends the Strauss exponent known for b = 0 . Our results improve the known ones for K (x) = μ | x | - b , μ ∈ C . For general potentials, we highlight the impact of the behavior at the origin and infinity on the allowed range of α . In the defocusing case, we prove decay estimates provided that the potential satisfies some rigidity-type condition which leads to a scattering result. We give also a new scattering criterion taking into account the potential K. [ABSTRACT FROM AUTHOR] |
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