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$%J. M. Mercier, Some results on semilinear systems on the unbounded space R3, Rev. Mat. Complut. 13 (2000), no. 1, 207229.%$

Some Results on Semilinear Systems on the Unbounded Space R3
Jean Marc MERCIER
École Polytechnique
Centre de Mathématiques Appliquées
U. A. CNRS 756
91128 Palaiseau Cedex — France

Received: April 27, 1999
Revised: January 31, 2000

ABSTRACT

We study in this paper the following systems, using standard tools devoted to the analysis of semilinear elliptic problems on  3
R  :

        {- Du -w2u = e1uv2,
R(e1,e2)        21      2     ei = ±1, wi  (-  R,  for i= 1,2.
         Dv - w2v = e2vu,

where u  , v  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 e1e2 = +1  cases. A first type of solution consists in a ground state of R(-1,-1)  , exhibited by variational arguments, whose structure is a finite energy perturbation of a non trivial constant solution of R(-1,-1)  . A second type consists in a radial, oscillating, asymptotically null at infinity solution in the e1e2 = +1  cases that we exhibit using eigenvalue comparison and ordinary differential equation type arguments.

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