Existence of stable standing waves and instability of standing waves to a class of quasilinear Schrodinger equations with potential

摘要

For a class of quasilinear Schrodinger equations with harmonic potential of the form i phi(t) - -Delta phi + |x|(2)phi - |phi|(p-1)phi -2(Delta|phi|(2))phi, t >= 0, x is an element of R(N), we prove firstly the existence of stable standing waves for 1 < p < 3 + 4/N and then study the instability of standing waves for 3 + 4/N <= p < 3N+2/N-2. Our results indicate that the quasilinear term (Delta|phi|(2))phi makes the standing waves more stable than their counterpart in the semilinear case, which is consistent with the physical phenomena and is in striking contrast with the classical semilinear Schrodinger equations with potential.