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We consider the focusing $L^2$-supercritical Schr\"odinger equation in theexterior of a smooth, compact, strictly convex obstacle. We construct asolution behaving asymptotically as a solitary waves on $R^3$, as large time.When the velocity of the solitary wave is high, the existence of such asolution can be proved by a classical fixed point argument. To constructsolutions with arbitrary nonzero velocity, we use a compactness argumentsimilar to the one that was introduced by F.Merle in 1990 to construct solutionof NLS blowing up at several blow-up point together with a topological argumentusing Brouwer's theorem to control the unstable direction of the linearizedoperator at soliton. These solutions are arbitrarily close to the scatteringthreshold given by a previous work of R.Killip, M.Visan and X.Zhang which isthe same as the one on whole Euclidean space.

Construction of solitary wave solution of the nonlinear focusing  schr\"odinger equation outside a strictly convex obstacle for the  $L^2$-supercritical case