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arXiv · 2312.07097

Asymptotically homogeneous solutions of the supercritical Lane-Emden system

Louis Dupaigne·Hatem Hajlaoui·Marius Ghergu
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Abstract

We consider the Lane-Emden system-$\Delta$u = |v| p-1 v,-$\Delta$v = |u| q-1 u in R d. When p $\ge$ q $\ge$ 1, it is known that there exists a positive radial stable solution (u, v) $\in$ C 2 (R d) if and only if d $\ge$ 11 and (p, q) lies on or above the so-called Joseph-Lundgren curve introduced in [5]. In this paper, we prove that for d $\le$ 10, there is no positive stable solution (or merely stable outside a compact set and (p, q) does not lie on the critical Sobolev hyperbola), while for d $\ge$ 11, the Joseph-Lundgren curve is indeed the dividing line for the existence of such solutions, if one assumes in addition that they are asymptotically homogeneous (see Definition 1 below). Most of our results are optimal improvements of previous works in the litterature.

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Louis Dupaigne, Hatem Hajlaoui, Marius Ghergu. 2023-12-12. Asymptotically homogeneous solutions of the supercritical Lane-Emden system. https://arxiv.org/abs/2312.07097

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