SMS scnews item created by Daniel Daners at Thu 23 Oct 2014 0950
Type: Seminar
Distribution: World
Expiry: 27 Oct 2014
Calendar1: 27 Oct 2014
CalLoc1: AGR Carslaw 829

PDE Seminar

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


Joshua Ching
University of Sydney
Mon 27 October 2014 2-3pm, Carslaw 829 (AGR)


For m (0, 2) and q > max{0, 1 - m}, we consider the nonlinear elliptic equation Δu = uq|u|m in Ω \{0}, where Ω denotes a domain in N (N 2) containing 0. We obtain a complete classification of the behaviour near 0 (as well as at if Ω = N) for all positive C1(Ω \{0}) distribution solutions, together with corresponding existence results. When Ω = N, any positive solution with a removable singularity at 0 must be constant. The origin is a removable singularity for all positive solutions if and only if Eq|E|mL1(B r(0)) for any small r > 0 (possible for N 3), where Br(0) denotes any ball centred at 0 with small radius r > 0 and E denotes the fundamental solution of the Laplacian. If Eq|E|m L1(B r(0)) for small r > 0, then any given positive solution has either removable singularity at 0, or lim |x|0u(x)E(x) (0,) or lim |x|0|x|ϑu(x) = λ, where ϑ and λ are precisely determined positive constants. When Ω = N, any positive solution is radially symmetric and non-increasing with (possibly any) non-negative limit at .

Check also the PDE Seminar page. Enquiries to Daniel Hauer or Daniel Daners.