Morse theory and a scalar field equation on compact surfaces. (English) Zbl 1175.53052

Malchiodi, Andrea

Adv. Differ. Equ. 13, No. 11-12, 1109-1129 (2008).

Summary: The aim of this paper is to study a nonlinear scalar field equation on a \(R\)-dimensional Riemannian manifold \(\Sigma\) via a Morse-theoretical approach, based on some of the methods in [Z. Djadli and A. Malchiodi, Existence of conformal metrics with constant \(Q\)-curvature, Ann. Math., to appear]. Employing these ingredients, we derive an alternative and direct proof (plus a clear interpretation) of a degree formula obtained in [W. Chen and C. S. Lin, Commun. Pure Appl. Math. 56, No. 12, 1667–1727 (2003; Zbl 1032.58010)], which used refined blow-up estimates from [J. Li, Commun. Math. Phys. 200, No.2, 421-444 (1999; Zbl 0928.35057)] and [C. C. Chen and C. S. Lin, Commun. Pure Appl. Math. 55, No. 6, 728–771 (2002; Zbl 1040.53046)]. Related results are derived for the prescribed Q-curvature equation on four manifolds.

MSC:

53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35B33	Critical exponents in context of PDEs

Keywords:

Kazdan-Warner problem; prescribed Q-curvature

Citations:

Zbl 1032.58010; Zbl 0928.35057; Zbl 1040.53046