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$%J. M. Mercier, Some results on semilinear systems on the unbounded space R³, Rev. Mat. Complut. 13 (2000), no. 1, 207–229.%$

We study in this paper the following systems, using standard tools devoted to the analysis of semilinear elliptic problems on :

where , hold for real valued functions. These systems do not admit any non trivial finite energy solution. However, we exhibit infinitely many non trivial radial solutions in the cases. A first type of solution consists in a ground state of , exhibited by variational arguments, whose structure is a finite energy perturbation of a non trivial constant solution of . A second type consists in a radial, oscillating, asymptotically null at infinity solution in the cases that we exhibit using eigenvalue comparison and ordinary differential equation type arguments.

1991 Mathematics Subject Classification: 65K10, 35J50, 35J45.