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A note on a Liouville-type result for a system of fourth-order equations in \(\mathbb R^N\). (English) Zbl 1068.35512

Summary: We consider the fourth order system \(\Delta^2 u =v^\alpha,\Delta^2 v =u^\beta\) in \(\mathbb{R}^N\), for \(N\geq 5\), with \(\alpha\geq 1\), \(\beta\geq 1\), where \(\Delta^2\) is the bi-Laplacian operator. For \(1/(\alpha +1) +1/(\beta +1)>(N-4)/N\) we prove the non-existence of non-negative, radial, smooth solutions. For \(\alpha,\beta\leq (N+4)/(N-4)\) we show the non-existence of non-negative smooth solutions.

MSC:

35J60 Nonlinear elliptic equations
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