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Blow-up solutions of Liouville’s equation and quasi-normality. (English) Zbl 1472.30015

Let \(D\) be a domain in the complex plane and \(C > 0\). Let \(\mathcal{F}_C \) be the set of all functions \(f\) meromorphic in \(D\) for which the spherical area of \( f(D)\) on the Riemann sphere is at most \(C \pi\). Then it is shown that \(\mathcal{F}_C \) is quasi-normal of order at most \( C\). In particular, for every sequence \(\{ f_m \} \) in \(\mathcal{F}_C \) (after taking a subsequence), there is an \( f \) in \(\mathcal{F}_C\) such that (1) or (2) below holds.
(1) \(\{ f_m \}\) converges locally uniformly in \(D\) to \( f \);
(2) There exists a finite nonempty set \(S \subset D\) with at most \(C\) points for which (2a) \(\{ f_m \}\) converges locally uniformly in \(D \backslash S\) to \(f\), and for each \(p\) in \(S\) there exists a sequence \(\{ z_m \}\) in \(D\) such that \(\{z_m \}\) converges to \(p\) and \(\{ f_m^{\#} (z_m) \}\) converges to \(+\infty\); and (2b) for each \(p\) in \(S\) there exists a real number \(\alpha_p \geq 1\) such that in the measure theoretic sense \[\frac{1}{\pi}(f_m^{\#})^2\text{ converges to } \sum_{p\in S}\alpha_p\delta_p+\frac{1}{\pi}(f^{\#})^2.\] The authors note that the above may be viewed as extending to all meromorphic functions in \(\mathcal{F}_C\) some well-known work of H. Brézis and F. Merle [Commun. Partial Differ. Equations 16, No. 8–9, 1223–1253 (1991; Zbl 0746.35006)] on solutions of \(-\Delta u =4e^{2u}\) for locally univalent meromorphic functions. In the comparison (2a) may be seen to correspond with “Bubbling”, while (2b) corresponds with “Mass Concentration” in the Brezis-Merle work. Section 2 of the current manuscript contains a lengthy set of remarks and questions (including open questions) regarding the above comparison, while Section 3 on quasi-normality observes a criterion of Montel and Valiron may be applied to obtain \(\mathcal{F}_C\) quasi-normal. Also introduced in Section 3 is an extension of the Montel-Valiron criterion for quasi-normality where exceptional values are replaced by exceptional functions allowed to depend on the individual members of the family.

MSC:

30D45 Normal functions of one complex variable, normal families
35J65 Nonlinear boundary value problems for linear elliptic equations
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination

Citations:

Zbl 0746.35006
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References:

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