## 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

Zbl 0746.35006
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