Vol. 13, No. 3, 2020

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Isolated singularities for semilinear elliptic systems with power-law nonlinearity

Marius Ghergu, Sunghan Kim and Henrik Shahgholian

Vol. 13 (2020), No. 3, 701–739
DOI: 10.2140/apde.2020.13.701
Abstract

We study the system Δu = |u|α1u with 1 < α n+2 n2, where u = (u1,,um), m 1, is a C2 nonnegative function that develops an isolated singularity in a domain of n , n 3. Due to the multiplicity of the components of u, we observe a new Pohozaev invariant different than the usual one in the scalar case. Aligned with the classical theory of the scalar equation, we classify the solutions on the whole space as well as the punctured space, and analyze the exact asymptotic behavior of local solutions around the isolated singularity. On a technical level, we adopt the method of moving spheres and the balanced-energy-type monotonicity functionals.

Keywords
elliptic system, isolated singularity, asymptotic behavior, Pohozaev invariant
Mathematical Subject Classification 2010
Primary: 35J61
Secondary: 35J75, 35B40, 35C20
Milestones
Received: 25 April 2018
Revised: 22 January 2019
Accepted: 13 March 2019
Published: 15 April 2020
Authors
Marius Ghergu
School of Mathematics and Statistics
University College Dublin
Dublin
Ireland
Institute of Mathematics Simion Stoilow of the Romanian Academy
Bucharest
Romania
Sunghan Kim
Department of Mathematical Sciences
Seoul National University
Seoul
South Korea
Henrik Shahgholian
Department of Mathematics
Royal Institute of Technology
Stockholm
Sweden