In this paper, we consider the following nonlinear coupled elliptic systems with continuous potentials:{-ε~2?u +(1 + δP(x))u = μ1 u^3+ βuv^2 in ?,-ε~2?v +(1 + δQ(x))v = μ2 v^3+ βu^2 v in ?,u > 0, v > 0 i...In this paper, we consider the following nonlinear coupled elliptic systems with continuous potentials:{-ε~2?u +(1 + δP(x))u = μ1 u^3+ βuv^2 in ?,-ε~2?v +(1 + δQ(x))v = μ2 v^3+ βu^2 v in ?,u > 0, v > 0 in ?,(?u)/(?v)=(?ν)/(?ν)=0on ??,(A_ε)where ? is a smooth bounded domain in R^N for N = 2, 3, δ, ε, μ_1 and μ_2 are positive parameters, β∈ R,P(x) and Q(x) are two smooth potentials defined on ?, the closure of ?. Due to Liapunov-Schmidt reduction method, we prove that(A_ε) has at least O(1/(ε| ln ε|)~N) synchronized and O(1/(ε| ln ε|)^(2 N)) segregated vector solutions for ε and δ small enough and some β∈ R. Moreover, for each m ∈(0, N) there exist synchronized and segregated vector solutions for(A_ε) with energies in the order of ε^(N-m). Our results extend the result of Lin et al.(2007) from the Lin-Ni-Takagi problem to the nonlinear Schr¨odinger elliptic systems with continuous potentials.展开更多
文摘In this paper, we consider the following nonlinear coupled elliptic systems with continuous potentials:{-ε~2?u +(1 + δP(x))u = μ1 u^3+ βuv^2 in ?,-ε~2?v +(1 + δQ(x))v = μ2 v^3+ βu^2 v in ?,u > 0, v > 0 in ?,(?u)/(?v)=(?ν)/(?ν)=0on ??,(A_ε)where ? is a smooth bounded domain in R^N for N = 2, 3, δ, ε, μ_1 and μ_2 are positive parameters, β∈ R,P(x) and Q(x) are two smooth potentials defined on ?, the closure of ?. Due to Liapunov-Schmidt reduction method, we prove that(A_ε) has at least O(1/(ε| ln ε|)~N) synchronized and O(1/(ε| ln ε|)^(2 N)) segregated vector solutions for ε and δ small enough and some β∈ R. Moreover, for each m ∈(0, N) there exist synchronized and segregated vector solutions for(A_ε) with energies in the order of ε^(N-m). Our results extend the result of Lin et al.(2007) from the Lin-Ni-Takagi problem to the nonlinear Schr¨odinger elliptic systems with continuous potentials.
