We consider the focusing $ L^2 $-supercritical Schrödinger equation in the exterior of a smooth, compact, strictly convex obstacle $ \Theta \subset \mathbb{R}^3 $. We construct a solution behaving asymptotically as a solitary wave on $ \mathbb{R}^3, $ for large times. When the velocity of the solitary wave is high, the existence of such a solution can be proved by a classical fixed point argument. To construct solutions with arbitrary nonzero velocity, we use a compactness argument similar to the one that was introduced by F.Merle in 1990 to construct solutions of the NLS equation blowing up at several points together with a topological argument using Brouwer's theorem to control the unstable direction of the linearized operator at the soliton. These solutions are arbitrarily close to the scattering threshold given by a previous work of R. Killip, M. Visan and X. Zhang, which is the same as the one on the whole Euclidean space given by S. Roundenko and J. Holmer in the radial case and by the previous authors with T. Duyckaerts in the non-radial case.

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非线性严格Schrödinger方程在凸障碍物外的孤立波解的构造。 \ begin {document} $ L ^ 2 $ \ end {document}-超临界情况

我们考虑在光滑，紧凑，严格凸的障碍物$ \ Theta \ subset \ mathbb {R} ^ 3 $外部的聚焦$ L ^ 2 $-超临界Schrödinger方程。我们构造一个渐近地表现为$ \ mathbb {R} ^ 3，$上的孤波的解。当孤立波的速度很高时，这种解决方案的存在可以用经典的不动点论证来证明。要构造具有任意非零速度的解，我们使用类似于F.Merle在1990年引入的紧致论来构造在多个点爆炸的NLS方程的解，并使用Brouwer定理来控制不稳定的拓扑论孤子上线性化算子的方向。这些解任意接近于R的先前工作给出的散射阈值。