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On a variational problem for radial solutions to extremal elliptic equations

Authors:	Orazio Arena Pasquale Buonocore

Institution:	(1) Dipartimento di Matematica e Applicazioni per l’Architettura, Universita degli Studi di Firenze, piazza Ghiberti, 27, 50122 Florence, Italy;(2) Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Universita degli Studi di Napoli Federico II, via Cintia, Complesso Universitario di Monte S. Angelo, 80126 Naples, Italy

Abstract:	On the ball \|x\| ≤ 1 of R ^m, m ≥ 2, a radial variational problem, related to a priori estimates for solutions to extremal elliptic equations with fixed ellipticity constant α is investigated. Such a problem has been studied and solved see Manselli Ann. Mat. Pura Appl. (IV), t. LXXXIX:31–54, 1971] in L ^p spaces, with p ≤ m. In this paper, we assume p > m and we prove the existence of a positive number α ₀ = α ₀(p,m) such that if $\alpha_0\leq\alpha\leq \frac{1}{m}$ there exists a smooth function maximizing the problem, whose representation is explicitly determined as in Manselli Ann. Mat. Pura Appl. (IV), t. LXXXIX:31–54, 1971] This fact is no longer true if 0 < α < α ₀.

Keywords:	Non-divergence elliptic operators Maximizing operators Radial solutions Variational methods
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