Instability of Solitary Waves for the Generalized Klein-Gordon-Hartree Equation

In the present paper we show that the ground state standing waves e(i omega t) u(x) for the generalized Klein-Gordon-Hartree equation psi(tt) - Delta psi + psi - (vertical bar x vertical bar(-(n-alpha)) * vertical bar psi vertical bar(-(n-alpha)) * vertical bar psi vertical bar(p-2) psi = 0, alpha is an element of(0, n), n >= 3 in R-n, are orbitally unstable in the domain of all frequencies 0 <= omega < 1 for the supercritical case and for frequencies, strictly less than a critical frequency omega(c) = root p-1/n+alpha+1p(n-1) in the subcritical case.