We consider the system of perturbed Schroedinger equations{-ε^2△φ＋α（x）φ=β（x）ψ＋Fψ（x,φ,ψ）-ε^2△ψ＋α（x）ψ=β（x）φ＋Fφ（x,φ,ψ）ω：=（φ,ψ）∈H^1（R^N,R^2）where N≥1, α and β are continuous...We consider the system of perturbed Schroedinger equations{-ε^2△φ＋α（x）φ=β（x）ψ＋Fψ（x,φ,ψ）-ε^2△ψ＋α（x）ψ=β（x）φ＋Fφ（x,φ,ψ）ω：=（φ,ψ）∈H^1（R^N,R^2）where N≥1, α and β are continuous real functions on R^N, and F ： R^N×R^2 → R is of class C^1. We assume that either F（x,ω） is super-quadratic and subcritical in ω∈R^2 or it is of the form ～1/P（x）｜ω｜^p ＋1/2^＊K（x）｜ω｜^2^＊ with p E （2,2^＊） and 2^＊ = 2N/（N-2）, the Sobolev critical exponent, P（x） and K（x） are positive bounded functions. Under proper conditions we show that the system has at least one nontrivial solution ωε provided ε≤ε; and for any m∈N, there are m pairs of solutions ωε provided that ε≤εm and that F（x, ω） is,in addition, even in ω. Here ε and ωε are sufficiently small positive numbers. Moreover, the energy of ωε tends to 0 as ε→0.展开更多
文摘We consider the system of perturbed Schroedinger equations{-ε^2△φ＋α（x）φ=β（x）ψ＋Fψ（x,φ,ψ）-ε^2△ψ＋α（x）ψ=β（x）φ＋Fφ（x,φ,ψ）ω：=（φ,ψ）∈H^1（R^N,R^2）where N≥1, α and β are continuous real functions on R^N, and F ： R^N×R^2 → R is of class C^1. We assume that either F（x,ω） is super-quadratic and subcritical in ω∈R^2 or it is of the form ～1/P（x）｜ω｜^p ＋1/2^＊K（x）｜ω｜^2^＊ with p E （2,2^＊） and 2^＊ = 2N/（N-2）, the Sobolev critical exponent, P（x） and K（x） are positive bounded functions. Under proper conditions we show that the system has at least one nontrivial solution ωε provided ε≤ε; and for any m∈N, there are m pairs of solutions ωε provided that ε≤εm and that F（x, ω） is,in addition, even in ω. Here ε and ωε are sufficiently small positive numbers. Moreover, the energy of ωε tends to 0 as ε→0.
