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A Liouville-type theorem for elliptic systems. (English) Zbl 0820.35042
The authors consider the system \(- \triangle u = v^ \alpha\), \(- \triangle v = u^ \beta\) in the whole of \(\mathbb{R}^ N\), \(N \geq 3\). The question is to determine for which values of the exponents \(\alpha\) and \(\beta\) the only nonnegative solution \((u,v)\) is the trivial one.
Theorem: If \(\alpha > 0, \beta > 0\) are such that \(\alpha, \beta \leq (N + 2)/(N - 2)\), but not both are equal to \((N + 2)/(N - 2)\), then the only nonnegative \(C^ 2\) solution of the system in the whole \(\mathbb{R}^ N\) is the trivial one. If \(\alpha = \beta = (N + 2)/(N - 2)\), then \(u\) and \(v\) are radially symmetric with respect to some point of \(\mathbb{R}^ N\).
Reviewer: O.John (Praha)

MSC:
35J45 Systems of elliptic equations, general (MSC2000)
35J60 Nonlinear elliptic equations
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