Compactness and the Palais–Smale property for critical Kirchhoff equations in closed manifolds

Emmanuel Hebey

Vol. 280 (2016), No. 1, 41–50

DOI: 10.2140/pjm.2016.280.41

Abstract

We prove the Palais–Smale property and the compactness of solutions for critical Kirchhoff equations using solely energy arguments in the situation where no sign assumption is made on the solutions. We then prove the existence of a mountain-pass solution to the equation, discuss its ground-states structure, and, in extreme cases, prove uniqueness of this solution.

Keywords

compactness, ground-states, Kirchhoff equation, mountain-pass solution, Palais–Smale property

Mathematical Subject Classification 2010

Primary: 58J05

Milestones

Received: 17 April 2015

Revised: 3 May 2015

Accepted: 7 May 2015

Published: 28 December 2015