Existence and classification of singular solutions to nonlinear elliptic equations with a gradient term

Ching, Joshua; Cîrstea, Florica

doi:10.2140/apde.2015.8.1931

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1948-206X (online)

ISSN 2157-5045 (print)

Author index

To appear

Other MSP journals

Existence and classification of singular solutions to nonlinear elliptic equations with a gradient term

Joshua Ching and Florica Cîrstea

Vol. 8 (2015), No. 8, 1931–1962

DOI: 10.2140/apde.2015.8.1931

Abstract

We completely classify the behaviour near $0$ , as well as at $\infty$ when $Ω = ℝ^{N}$ , of all positive solutions of $Δ u = u^{q} | \nabla u |^{m}$ in $Ω ∖ {0}$ , where $Ω$ is a domain in $ℝ^{N}$ ( $N \geq 2$ ) and $0 \in Ω$ . Here, $q \geq 0$ and $m \in (0, 2)$ satisfy $m + q > 1$ . Our classification depends on the position of $q$ relative to the critical exponent $q_{*} : = (N - m (N - 1)) ∕ (N - 2)$ (with $q_{*} = \infty$ if $N = 2$ ). We prove the following: if $q < q_{*}$ , then any positive solution $u$ has either (1) a removable singularity at $0$ , or (2) a weak singularity at $0$ ( $lim_{| x | \to 0} u (x) ∕ E (x) \in (0, \infty)$ , where $E$ denotes the fundamental solution of the Laplacian), or (3) $lim_{| x | \to 0} | x |^{ϑ} u (x) = λ$ , where $ϑ$ and $λ$ are uniquely determined positive constants (a strong singularity). If $q \geq q_{*}$ (for $N > 2$ ), then $0$ is a removable singularity for all positive solutions. Furthermore, for any positive solution in $ℝ^{N} ∖ {0}$ , we show that it is either constant or has a nonremovable singularity at $0$ (weak or strong). The latter case is possible only for $q < q_{*}$ , where we use a new iteration technique to prove that all positive solutions are radial, nonincreasing and converging to any nonnegative number at $\infty$ . This is in sharp contrast to the case of $m = 0$ and $q > 1$ , when all solutions decay to $0$ . Our classification theorems are accompanied by corresponding existence results in which we emphasise the more difficult case of $m \in (0, 1)$ , where new phenomena arise.

Keywords

nonlinear elliptic equations, isolated singularities, Leray–Schauder fixed point theorem, Liouville-type result

Mathematical Subject Classification 2010

Primary: 35J25

Secondary: 35B40, 35J60

Milestones

Received: 17 February 2015

Revised: 22 July 2015

Accepted: 7 September 2015

Published: 23 December 2015

Authors

Joshua Ching
School of Mathematics and Statistics The University of Sydney Sydney NSW 2006 Australia

Florica Cîrstea
School of Mathematics and Statistics The University of Sydney Sydney NSW 2006 Australia

Vol. 8, No. 8, 2015