(untitled) (Journal of Partial Differential Equations, 2006)
We consider the system of perturbed Schrödinger equations -ε²Δφ + α(x)φ = β(x)ψ + F_ψ(x, φ, ψ) -ε²Δψ + α(x)ψ = β(x)φ + F_φ(x, φ, ψ) w := (φ, ψ) ∈ H¹(R^N, R²) where N ≥ 1, α and β are continuous real functions on R^N, and F : R^N × R² → R is of class C¹. We assume that either F(x, w) is super-quadratic and subcritical in w ∈ R² or it is of the form ∼ 1pP(x)|w|^p + 12∗K(x)|w|2∗ with p ∈ (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 w_ε provided ε ≤ ξ and for any m ∈ N, there are m pairs of solutions w_ε provided that ε ≤ ξ_m and that F(x, w) is, in addition, even in w. Here ξ and ξ_m are sufficiently small positive numbers. Moreover, the energy of w_ε tends to 0 as ε → 0.