Semiclassical States of Hamiltonian System of Schrodinger Equations with Subcritical and Critical Nonlinearities

Abstract

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(\mathbb{R}^N, \mathbb{R}²) where N ≥ 1, α and β are continuous real functions on \mathbb{R}^N, and F : \mathbb{R}^N × \mathbb{R}² → \mathbb{R} is of class C¹. We assume that either F(x, w) is super-quadratic and subcritical in w ∈ \mathbb{R}² or it is of the form ∼ \frac{1}{p}P(x)|w|^p + \frac{1}{2∗}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 ∈ \mathbb{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.