Linking in infinite dimensional spaces using the Leray-Schauder index

Abstract

Many problems arising in science and engineering call for the solving of the Euler-Lagrange equations of functionals. Thus, solving the Euler-Lagrange equations is tantamount to finding critical points of the corresponding functional. An idea that has been very successful is to find appropriate sets that separate the functional. This method is called linking. Two sets A,B are said to separate a functional G if the supremum of G on A is less than or equal to the infimum of G on B. Two sets of the space are said to link if they produce a critical sequence whenever they separate a functional. If the critical sequence has a convergent sub-sequence, then that produces a critical point. Finding sets that separate a functional is quite easy, but determining whether or not the sets link is quite another story. It appears that the only way we can check to see if two sets link, is to require that one of them be contained in a finite-dimensional subspace. The reason is that in order to verify the definition, we need to invoke the Brouwer fixed point theorem. Our aim is to find a counterpart of linking that holds true when both sets are infinite dimensional. We adjust our definitions to accommodate infinite dimensions. These definitions reduced to the usual definitions when one set is finite dimensional. In order to prove the corresponding theorems, we make adjustments to the topology of the space and introduce infinite dimensional splitting. This allows us to use a form of compactness on infinite dimensional subspaces which does not exist under the usual topology. We lose the Brouwer index, but we are able to replace it with the Leray-Schauder index. We carry out the details in Sections 5 and 6. In Section 7 we solve a system of equations which require infinite dimensional splitting.