Domain variation for certain sets of solutions and applications
Edward Norman Dancer · Topological Methods in Nonlinear Analysis · 1996
Dedicated to Louis Nirenberg on the occasion of his 70th birthday The purpose of this paper is three-fold. We generalize work of our earlier papers [8]–[10] to show that certain solutions or sets of solutions of (1) −∆u = f(u) in Ω (or systems of equations) with either Dirichlet or Neumann boundary conditions continue if Ω is perturbed in quite a general way. More precisely, in the earlier work, we showed that if the set of solutions has non-zero Leray–Schauder degree, then it does continue if Ω is perturbed. Here we prove similar results when we consider sets of solutions of non-zero homotopy index (or Morse numbers), where the homotopy index is defined in Rybakowski [22]. The proof of this is much more delicate than the earlier case since we need to retain the variational structure. We become interested in this problem for two reasons. Firstly, if Ω is invariant under the orthogonal action of a compact Lie group G, then the set of solutions of (1) is invariant ynder the natural action of the symmetry group G. Thusthe solutions of (1) are usually orbits under this group action rather than isolated points. Then a theorem of Sylvester [25] implies that these orbits frequently have Leray–Schauder degree zero (for example if G = S1 and the orbit consists of more than one point). Thus the old arguments do not apply but the new result does apply. Note that one cannot always avoid the problem by using subspaces