Isolated singularities for semilinear elliptic systems with power-law nonlinearity

Ghergu, Marius; Kim, Sunghan; Shahgholian, Henrik

doi:10.2140/apde.2020.13.701

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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 $- Δ u = | u |^{α - 1} u$ with $1 < α \leq \frac{n + 2}{n - 2}$ , where $u = (u_{1}, \dots, u_{m})$ , $m \geq 1$ , is a $C^{2}$ nonnegative function that develops an isolated singularity in a domain of $ℝ^{n}$ , $n \geq 3$ . 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

Vol. 13, No. 3, 2020