A blow‐up result for the travelling waves of the pseudo‐relativistic Hartree equation with small velocity

Abstract

In this paper, we consider the pseudo‐relativistic Hartree equation and study travelling solitary waves of the form ψ(t, x) = eitμφ(x − v t) , where denotes travelling velocity. Fröhlich, Jonsson and Lenzmann in [Comm. Math. Phys. 2007, 274:1‐30] proved that for |v|<1 there exists a critical constant Nc(v), such that the travelling waves exist if and only if 0 < N < Nc(v), where N denotes particle number. In this paper, we consider with 0 < β < 1, and let . We find that Nc(β) is Lipschitz continuity with respect to β. Based on this fact, we then prove that the boosted ground states φβ with satisfy . The explicit blow‐up profile and rate will be computed.