On the Orbital Stability for a Class of Nonautonomous Nls

Following the original approach introduced by T. Cazenave and P.L. Lions in [4] we prove the existence and the orbital stability of standing waves for the following class of NLS: (0.1) i∂tu + ∆u − V (x)u + Q(x)u|u| p−2 = 0, (t, x) ∈ R × R n , 2 < p < 2 + 4 n and (0.2) i∂tu − ∆ 2 u − V (x)u + Q(x)u|u| p−2 = 0, (t, x) ∈ R × R n , 2 < p < 2 + 8 n under suitable assumptions on the potentials V (x) and Q(x). More precisely we assume V (x), Q(x) ∈ L ∞ (R n) and meas{Q(x) > λ 0 } ∈ (0, ∞) for a suitable λ 0 > 0. The main point is the analysis of the compactness of minimiziang sequences to suitable constrained minimization problems related to (0.1) and (0.2).