Asymptotically homogeneous solutions of the supercritical Lane-Emeden system

Solutions asymptotiquement homogènes du système de Lane-Emden surcritique

Louis Dupaigne (1, 2, 3) , Hatem Hajlaoui (4) , Marius Ghergu (5)

1 ICJ - Institut Camille Jordan
2 EDPA - Équations aux dérivées partielles, analyse
3 UCBL - Université Claude Bernard Lyon 1
4 ISSAT Kairouan - Institut Supérieur des Sciences Appliquées et de Technologie de Kairouan [Kairouan]
5 UCD - University College Dublin [Dublin]

We consider the Lane-Emden system-∆u = |v| p-1 v,-∆v = |u| q-1 u in R d. When p ≥ q ≥ 1, it is known that there exists a positive radial stable solution (u, v) ∈ C 2 (R d) if and only if d ≥ 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 ≤ 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 ≥ 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.

Domaines: Equations aux dérivées partielles [math.AP]

Louis Dupaigne, Hatem Hajlaoui, Marius Ghergu. Asymptotically homogeneous solutions of the supercritical Lane-Emeden system. 2024. ⟨hal-04335633v3⟩