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SEMICLASSICAL SOLUTIONS FOR LINEARLY COUPLED SCHR¨ODINGERSCHR¨ SCHR¨ODINGER EQUATIONS

2014
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Electronic Journal of Differential Equations
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unpublished

We consider the system of coupled nonlinear Schrödinger equations −ε 2 ∆u + a(x)u = Hu(x, u, v) + µ(x)v, x ∈ R N , −ε 2 ∆v + b(x)v = Hv(x, u, v) + µ(x)u, x ∈ R N , u, v ∈ H 1 (R N), where N ≥ 3, a, b, µ ∈ C(R N) and Hu, Hv ∈ C(R N × R 2 , R). Under conditions that a 0 = inf a = 0 or b 0 = inf b = 0 and |µ(x)| 2 ≤ θa(x)b(x) with θ ∈ (0, 1) and some mild assumptions on H, we show that the system has at least one nontrivial solution provided that 0 < ε ≤ ε 0 , where the bound ε 0 is formulated in terms of N, a, b and H.

fatcat:nekalhph4fb2dl7tcwolmcavq4