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Asymptotically homogeneous solutions of the supercritical Lane-Emden system

Louis Dupaigne, Hatem Hajlaoui, Marius Ghergu

Abstract

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.

Asymptotically homogeneous solutions of the supercritical Lane-Emden system

Abstract

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.
Paper Structure (7 sections, 22 theorems, 150 equations)

This paper contains 7 sections, 22 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Stability revisited
  3. The case $\alpha+\beta\geq d-4$
  4. Fast decay solutions
  5. Stable homogeneous solutions
  6. The case $d-4>\alpha+\beta$: an iteration method in the spirit of De Giorgi
  7. The case $d-4>\alpha+\beta$: asymptotics of solutions stable outside a compact set

Key Result

Theorem 1

Let $d\leq 10$ and $p\geq q\geq1$ be such that $pq>1$. If $(u,v)\in C^2(\mathbb R^d)$ is a nonnegative solution which is stable, or merely stable outside a compact set but with then $u=v=0$.

Theorems & Definitions (40)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Remark 1
  • Lemma 1
  • proof
  • ...and 30 more