Schrödinger equations with a spatially decaying nonlinearity: existence and stability of standing waves
نویسندگان
چکیده
For N ≥ 3 and p > 1, we consider the nonlinear Schrödinger equation i∂tw +∆xw + V (x) |w|p−1 w = 0 where w = w(t, x) : R× R → C with a potential V that decays at infinity like |x|−b for some b ∈ (0, 2). A standing wave is a solution of the form w(t, x) = eu(x) where λ > 0 and u : R → R. For 1 < p < 1 + (4− 2b)/(N − 2), we establish the existence of a C1-branch of standing waves parametrized by frequencies λ in a right neighbourhood of 0. We also prove that these standing waves are orbitally stable if 1 < p < 1 + (4− 2b)/N and unstable if 1 + (4− 2b)/N < p < 1 + (4− 2b)/(N − 2).
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