Stationary Kirchhoff systems in closed high dimensional manifolds

Abstract

We discuss existence of solutions, compactness and stability properties for Kirchhoff-type systems in closed [Formula: see text]-manifolds [Formula: see text], [Formula: see text]. The Kirchhoff systems we consider are written as [Formula: see text] for all [Formula: see text], where [Formula: see text] is the Laplace–Beltrami operator, [Formula: see text] is a [Formula: see text]-map from [Formula: see text] into the space [Formula: see text] of symmetric [Formula: see text] matrices with real entries, the [Formula: see text]’s are the components of [Formula: see text], [Formula: see text], [Formula: see text] is the Euclidean norm of [Formula: see text], [Formula: see text] is the critical Sobolev exponent, and we require that [Formula: see text] in [Formula: see text] for all [Formula: see text].

Klein–Gordon–Maxwell–Proca type systems in the electro-magneto-static case: the high dimensional case

“…Concerning the notations in (6.54), the T i,ε 's terms may matter, while the R i,ε 's terms will eventually be only remainder terms. As in Hebey and Thizy [20,Lemma 9.4] we get from (6.27)-(6.28), using (6.5), that…”

“…At that stage, we can conclude the proof of Theorem 6.1 following that of Hebey and Thizy [20,Theorem 8.1].…”

“…(6.9) Focusing on the first equation in (6.1), and using (6.5) and (6.8), as a by product of the classification in Caffarelli, Gidas and Spruck [5], we get as in [9,20]…”

“…. By the Pohozaev identity as in Druet, Hebey and Vétois [11] (see also Hebey [19] and Hebey and Thizy [20]):…”

“…This proves (6.74). Assuming (6.74), the proof of (6.75) is by now rather standard (see for instance Hebey and Thizy [20,Lemma 9.6]). It uses in particular (6.5), the first equation in (6.1) and the Bôcher's theorem about nonnegative harmonic functions.…”