The Threshold Of Blow-up Solutions And Strong Instability For The Standing Wave Of Davey-Stewartson Systems

The aim of this paper is to study the generalized Davey-Stewartsonequationwhere a > 0, 1 <p< n+2/(n-2)+n?{2,3}.When a = 1,p = 3 and n = 3, Li, Zhang et al. [J. Differential Equation-s. 250, 2197-2226 (2011).] under the condition of M[?]H[?] > M[u0]H[u0],obtained some sharp criteria for the blowup solutions of this equation. In this paper, we consider the complementary case M[?]H[?] ? M[u0]H[u0] ? C (C > 0 is a constant), deriving a new threshold of blow-up solutions.When a > 0,n ? {2,3} and 1+4/n?p<n+2/(n-2)+, Gan, Zhang [Commun.Math. Phys. 283, 93-125 (2008)] obtained the strongly instability of the ground state standing waves, under a assumption about the frequency of the standing wave. In the present paper, We establish the same conclusion without that as-sumption.