Variational aspects of the Seiberg-Witten functional (Q1913316)

\textit{N. Seiberg} and \textit{E. Witten} [Nucl. Phys., B 426, No. 1, 19-52 (1994)] introduced a new monopole equation which yields new differential-topological invariants of 4-dimensional manifolds, closely related to Donaldson's polynomial invariants. The authors define the Seiberg-Witten functional and consider the corresponding Euler-Lagrange equations. It is shown that their weak solutions are smooth. The main result is that this functional satisfies the Palais-Smale condition. As a consequence, the existence of unstable solutions is proved, which can be used to define more invariants for 4-manifolds.