In this paper, we give a simpler proof for Ohta's theorems [1995, Ann. Inst. Henri Poincare, 63, 111; 1995, Diff. Integral Eq., 8, 1775] on the strong instability of the ground states for a generalized Davey-Stewartson system. In addition, a sufficient condition is given to ensure the nonexistence of a minimizer for a variational problem, which is related to the stability of the standing waves of the Davey-Stewartson system. This result shows that the stability result of Ohta [Diff. Integral Eq., 8, 1775] is sharp.
Yuanping DENG
,
Xiaoguan LI
,
Qian SHENG
. ON THE INSTABILITY OF GROUND STATES FOR A GENERALIZED DAVEY-STEWARTSON SYSTEM[J]. Acta mathematica scientia, Series B, 2020
, 40(4)
: 1081
-1090
.
DOI: 10.1007/s10473-020-0414-0
[1] Berestycki H, Cazenave T. Instabilite des etats stationnaires dans les equations de Schrödinger et de KleinGordon non linearires. C R Acad Sci Paris, Seire I, 1981, 293:489-492
[2] Cazenave T. Semilinear Schrödinger equations//Courant Lecture Notes in Mathematics, vol. 10. New York:New York University, Courant Institute of Mathematical Sciences; Providence, RI:American Mathematical Society, 2003
[3] Cipolatti R. On the existence of standing waves for a Davey-Stewartson system. Comm Ppartial Differ Equ, 1992, 17:967-988
[4] Cipolatti R. On the instability of ground states for a Davey-Stewartson system. Ann Inst Henri Poincaré Phys Théory, 1993, 58:85-104
[5] Davey A, Stewartson K. On three-dimensional packets of surface waves. Proc R Soc London A, 1974, 338:101-110
[6] Gan Z, Zhang J. Sharp threshold of global existence and instability of standing wave for a Davey-Stewartson system. Comm Math Phys, 2008, 283:93-125
[7] Ghidaglia J M, Saut J C. On the initial value problem for the Davey-Stewartson systems. Nonlinearity, 1990, 3:475-506
[8] Guo B L, Wang B X. The Cauchy problem for Davey-Stewartson systems. Comm Pure Appl Math, 199952:1477-1490
[9] Li Y, Guo B L. Existence and decay of weak solutions to degenerate Davey-Stewartson equaitons. Acta Math Sci, 2002, 22B(3):302-310
[10] Ohta M. Stability of standing waves for the generalized Davey-Stewartson system. J Dyn Differ Eqs, 1994, 6:325-334
[11] Ohta M. Instability of standing waves for the generalized Davey-Stewartson system. Ann Inst Henri Poincaré, 1995, 62:69-80
[12] Ohta M. Blow-up solutions and strong instability of standing waves for the generalized Davey-Stewartson system in $\mathbb{R}^2$. Ann Inst Henri Poincare, 1995, 63:111-117
[13] Ohta M. Stability and instability of standing waves for the generalized Davey-Stewartson system. Differ Integral Equ, 1995, 8:1775-1788
