This paper is devoted to study the following systems of coupled elliptic equations with quadratic nonlinearity −ε 2 ∆v + P (x)v = µvw, x ∈ R N , −ε 2 ∆w + Q(x)w = µ 2 v 2 + γw 2 , x ∈ R N , which arises from second-harmonic generation in quadratic optical media. We assume that the potential functions P (x) and Q(x) are positive functions and have a strict local maxima at x 0. Applying the finite dimensional reduction method, for any integer 1 ≤ k ≤ N + 1, we prove the existence of positive solutions which have k local maximum points that concentrate at x 0 simultaneously when ε is small.
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