Vol. 8, No. 8, 2015

Download this article
Download this article For screen
For printing
Recent Issues

Volume 10
Issue 2, 253–512
Issue 1, 1–252

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
Cover
About the Cover
Editorial Board
Editors’ Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Subscriptions
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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
Abstract

We completely classify the behaviour near 0, as well as at when Ω = N, of all positive solutions of Δu = uq|u|m in Ω {0}, where Ω is a domain in N (N 2) and 0 Ω. Here, q 0 and m (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 = 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|0u(x)E(x) (0,), where E denotes the fundamental solution of the Laplacian), or (3)  lim|x|0|x|ϑu(x) = λ, where ϑ and λ are uniquely determined positive constants (a strong singularity). If q q (for N > 2), then 0 is a removable singularity for all positive solutions. Furthermore, for any positive solution in 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 . 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 (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