Asymptotically homogeneous solutions of the supercritical Lane-Emden system
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| Format: | Preprint |
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2023
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| author | Dupaigne, Louis Hajlaoui, Hatem Ghergu, Marius |
| author_facet | Dupaigne, Louis Hajlaoui, Hatem Ghergu, Marius |
| contents | We consider the Lane-Emden system-$Δ$u = |v| p-1 v,-$Δ$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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_07097 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Asymptotically homogeneous solutions of the supercritical Lane-Emden system Dupaigne, Louis Hajlaoui, Hatem Ghergu, Marius Analysis of PDEs We consider the Lane-Emden system-$Δ$u = |v| p-1 v,-$Δ$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. |
| title | Asymptotically homogeneous solutions of the supercritical Lane-Emden system |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2312.07097 |