Existence of finite energy solutions for semi-linear elliptic systems

We show that the only solution of R (epsilon(1), epsilon(2)) which are semi-linear elliptic systems treated in the full space R(3) for real valued (u, v) functions, is the trivial one. However, we show that the R(-1, -1) system admits solutions that are non trivial finite energy perturbations of the particular (a, 0) solution, with a not equal 0.