Regularity of the extremal solutions for the Liouville system
Résumé
In this short note, we study the smoothness of the extremal solutions to the following system of equations: (1) −∆u = µe v in Ω, −∆v = λe u in Ω, u = v = 0 on ∂Ω, where λ, µ > 0 are parameters and Ω is a smoothly bounded domain of R N , N ≥ 1. As shown by M. Montenegro (see [6]), there exists a limiting curve Υ in the first quadrant of the (λ, µ)-plane serving as borderline for existence of classical solutions of (1). He also proved the existence of a weak solution u * for every (λ * , µ *) on the curve Υ and left open the question of its regularity. Following standard terminology (see e.g. the books [3], [5] for an introduction to this vast subject), u * is called an extremal solution. Our result is the following. Theorem 1 Let 1 ≤ N ≤ 9. Then, extremal solutions to (1) are smooth.
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