Multi-bump bound states for a nonlinear Schrödinger system with electromagnetic fields

In this paper, we are concerned with the following nonlinear Schrödinger system with electromagnetic fields equation(Sλ){−(∇+iA(x))2u(x)+λV(x)u(x)=2αα+β|u(x)|α−2|v(x)|βu(x),x∈RN,−(∇+iB(x))2v(x)+λW(x)v(x)=2βα+β|u(x)|α|v(x)|β−2v(x),x∈RN,|u(x)|→0,|v(x)|→0as |x|→∞ for sufficiently large λλ, where i is the imaginary unit, α>1,β>1,α+β<Θα>1,β>1,α+β<Θ and Θ=2NN−2 for N≥3,Θ=+∞N≥3,Θ=+∞ for N=1,2N=1,2. A(x)A(x) and B(x)B(x) are real-valued electromagnetic vector potentials. V(x)V(x) and W(x)W(x) are real-valued continuous nonnegative functions on RNRN. By modifying the nonlinearity and using the decay flow we show that if Ω:=int V−1(0)∩int W−1(0) has several isolated connected components Ω1,Ω2,…,ΩkΩ1,Ω2,…,Ωk such that the interior of ΩiΩi is not empty and ∂Ωi∂Ωi is smooth for all i∈{1,2,…,k}i∈{1,2,…,k}, then for any non-empty subset J⊂{1,2,…,k}J⊂{1,2,…,k} there exists a solution (uλ,vλ)(uλ,vλ) of (Sλ)(Sλ) for λ>0λ>0 large. Moreover for any sequence λn→∞λn→∞, up to a subsequence (uλn,vλn)(uλn,vλn) converges in Ωj(j∈J) to a least energy solution of the following limit problem equation(DΩj){−(∇+iA(x))2u(x)=2αα+β|u(x)|α−2|v(x)|βu(x),x∈Ωj,−(∇+iB(x))2v(x)=2βα+β|u(x)|α|v(x)|β−2v(x),x∈Ωj,(u(x),v(x))∈HA,B0,1(Ωj) and outside of ⋃j∈JΩj⋃j∈JΩj to (0,0)(0,0).