Communications in Mathematical Sciences

Volume 19 (2021)

Number 2

Singular solutions to some semilinear elliptic equations: an approach of Born–Infeld approximation

Pages: 557 – 584

DOI: https://dx.doi.org/10.4310/CMS.2021.v19.n2.a11

Authors

Chia-Yu Hsieh (Department of Mathematics, Chinese University of Hong Kong, Sha Tin, N.T., Hong Kong)

Ho-Man Tai (Department of Mathematics, Chinese University of Hong Kong, Sha Tin, N.T., Hong Kong)

Yong Yu (Department of Mathematics, Chinese University of Hong Kong, Sha Tin, N.T., Hong Kong)

Abstract

We construct singular solutions to a semilinear elliptic equation with exponential nonlinearity on $\Omega \subset \mathbb{R}^2$ by a shrinking hole argument, which we call Born–Infeld approximation scheme. With some natural assumptions on the nonlinearity $f (e^u)$, we classify all these singular solutions by the order of $u$ near each singularity. To show the existence of singular solutions, we introduce an inverse problem and utilize a Born–Infeld approximation scheme. Ruling out a possible occurrence of bubbling phenomenon, we show that as the Born–Infeld parameter tends to infinity, solutions of the inverse problem on a subdomain of $\Omega$ with finitely many holes can be used to approximate singular solutions to the original equation, provided that the size of each hole is carefully given and satisfies some constraint condition on the total flux. Our work rigorously justifies the Born–Infeld–Higgs approximation to the Abelian Maxwell–Higgs theory. When $b=1$, we also find finite-energy solutions to the equation of Born–Infeld gauged harmonic maps, which have finitely many magnetic singularities in $\Omega$.

Keywords

semilinear elliptic equation, singular solution, Born–Infeld approximation, inverse problem

2010 Mathematics Subject Classification

35J61, 35J75, 35R30

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Y. Yu is partially supported by RGC grants of Hong Kong (14306217, 14302718).

Received 12 February 2020

Accepted 21 September 2020

Published 12 April 2021