Nonlinear Fractional Schrödinger Equations coupled by power--type nonlinearities

Abstract

In this work we study the following class of systems of coupled nonlinear fractional nonlinear Schrödinger equations, { (−∆)u1 + λ1u1 = μ1|u1|u1 + β|u2||u1|u1 in R , (−∆)u2 + λ2u2 = μ2|u2|u2 + β|u1||u2|u2 in R , where u1, u2 ∈ W (R ), with N = 1, 2, 3; λj , μj > 0, j = 1, 2, β ∈ R, p ≥ 2 and p− 1 2p N < s < 1. Precisely, we prove the existence of positive radial bound and ground state solutions provided the parameters β, p, λj , μj , (j = 1, 2) satisfy appropriate conditions. We also study the previous system with m-equations, (−∆)uj + λjuj = μj |uj |uj + m ∑ k=1 k 6=j βjk|uk||uj |uj , uj ∈W (R ); j = 1, . . . ,m where λj , μj > 0 for j = 1, . . . ,m ≥ 3, the coupling parameters βjk = βkj ∈ R for j, k = 1, . . . ,m, j 6= k. For this system we prove similar results as for m = 2, depending on the values of the parameters βjk, p, λj , μj , (for j, k = 1, . . . ,m, j 6= k).