Vol. 8, No. 8, 2015

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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

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.

nonlinear elliptic equations, isolated singularities, Leray–Schauder fixed point theorem, Liouville-type result
Mathematical Subject Classification 2010
Primary: 35J25
Secondary: 35B40, 35J60
Received: 17 February 2015
Revised: 22 July 2015
Accepted: 7 September 2015
Published: 23 December 2015
Joshua Ching
School of Mathematics and Statistics
The University of Sydney
Sydney NSW 2006
Florica Cîrstea
School of Mathematics and Statistics
The University of Sydney
Sydney NSW 2006