Abstract

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) 2 R × Rⁿ, 2 < p < 2 + 4/n and
(0.2)
i@_tu − Δ²u − V (x)u + Q(x)u|u|^p−2 = 0, (t, x) 2 R × Rⁿ, 2 < p < 2 + 8/n under suitable assumptions on the potentials V (x) and Q(x). More precisely we assume V (x),Q(x) 2 L∞(Rⁿ) and meas{Q(x) < λ₀} ε (0,∞) for a suitable λ₀<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).

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