STABILITY OF STANDING WAVES FOR NONLINEAR SCHRODINGER EQUATIONS WITH CRITICAL POWER NONLINEARITY AND POTENTIALS

We study the stability of standing waves e(iwt)phi(omega)(x) for a nonlinear Schrodinger equation with critical power nonlinearity vertical bar u vertical bar(4/n)u and a potential V/(x) in R-n. Here, omega is an element of R and phi(omega)(x) is a ground state of the stationary problem. Under suitable assumptions on V(x), we show that e(iwt)phi(omega)(x) is stable for sufficiently large omega. This result gives a different phenomenon from the case V(x) equivalent to 0 where the strong instability was proved by M. I. Weinstein [25].