Advances in Differential Equations

Morse theory and a scalar field equation on compact surfaces

Andrea Malchiodi

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The aim of this paper is to study a nonlinear scalar field equation on a surface $\Sigma$ via a Morse-theoretical approach, based on some of the methods in [25]. Employing these ingredients, we derive an alternative and direct proof (plus a clear interpretation) of a degree formula obtained in [18], which used refined blow-up estimates from [34] and [17]. Related results are derived for the prescribed $Q$-curvature equation on four manifolds.

Article information

Adv. Differential Equations, Volume 13, Number 11-12 (2008), 1109-1129.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]
Secondary: 35B33: Critical exponents 35J60: Nonlinear elliptic equations 53A30: Conformal differential geometry


Malchiodi, Andrea. Morse theory and a scalar field equation on compact surfaces. Adv. Differential Equations 13 (2008), no. 11-12, 1109--1129.

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