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

Abstract

Let Ω be a domain in R with N ≥ 2 and 0 ∈ Ω. For 0 0 and m+q > 1, we obtain a complete classification of the behaviour near 0 (as well at∞ if Ω = R ) for all positive C(Ω\{0}) solutions of the elliptic equation ∆u = u|∇u| in Ω {0}, together with corresponding existence results. We prove that (a) when Ω = R , any positive solution with a removable singularity at 0 must be constant; (b) If q∗ := N−m (N−1) N−2 for N ≥ 2 and E denotes the fundamental solution of the Laplacian, then for 0 ≤ q < q∗, any positive solution has either a removable singularity at 0, or lim|x|→0 u(x)/E(x) ∈ (0,∞) or lim|x|→0 |x|u(x) = λ with θ and λ uniquely determined positive constants. When Ω = R , we establish that any positive solution is radially symmetric and non-increasing with (possibly any) non-negative limit at ∞. (c) If, in turn, q ≥ q∗ for N ≥ 3, then 0 is a removable singularity for all positive solutions. This is joint work with Joshua Ching (The University of Sydney). School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia