An implicit function theorem with symmetries and its application to nonlinear eigenvalue equations

Abstract

In this paper we prove a G-invariant implicit function theorem and indicate how it can be used to improve an earlier result of the author on the bifurcation of solutions of nonlinear equations in the presence of continuous groups of symmetries. We also use our theorem to show that, under reasonable hypotheses, the method of looking for solutions in invariant subspaces yields all solutions. This can be used to answer a question raised by Sattinger [J. Math. Phys. 19 (1978), 1729]. The abstract result is also of interest because it provides a theorem which should be of use in other symmetric situations.