Existence of self-dual non-topological solutions in the Chern–Simons Higgs model

Choe, Kwangseok; Kim, Namkwon; Lin, Chang-Shou

Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 6, pp. 837-852.

Lʼobjectif de cet article est de prouver lʼexistence de solutions non-topologiques du modèle de Chern–Simons Higgs dans ${ℝ}^2$. Un problème de longue date existe pour cette équation : Soit N points vortex et $\beta >8\pi (N+1)$, existe-t-il une solution non-topologique dans ${ℝ}^2$ telle que le flux magnétique total est égal à $\beta /2$ ? Dans cet article, nous prouvons lʼexistence dʼune solution pour $\beta \notin \left\{8\pi N\frac{k}{k-1}\right|k=2,\cdots ,N\}$. Nous appliquons lʼanalyse par bulles et la theorie de Leray–Schauder pour résoudre ce problème.

In this paper we investigate the existence of non-topological solutions of the Chern–Simons Higgs model in ${ℝ}^2$. A long standing problem for this equation is: Given N vortex points and $\beta >8\pi (N+1)$, does there exist a non-topological solution in ${ℝ}^2$ such that the total magnetic flux is equal to $\beta /2$? In this paper, we prove the existence of such a solution if $\beta \notin \left\{8\pi N\frac{k}{k-1}\right|k=2,\cdots ,N\}$. We apply the bubbling analysis and the Leray–Schauder degree theory to solve this problem.

Mots clés : Semi-linear PDE, Non-topological vortices, Chern–Simons Higgs model

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