Critical Point Theory in Infinite Dimensional Spaces Using the Leray–Schauder Index

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 sandwich the functional. This means that the functional is bounded from above on one of the sets and bounded from below on the other. Two sets of the space are said to form a sandwich if they produce a critical sequence whenever they sandwich a functional. If the critical sequence has a convergent subsequence, then that produces a critical point. Finding sets that sandwich a functional is quite easy, but determining whether or not the sets form a sandwich is quite another story. It appears that the only way we can check to see if two sets form a sandwich 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 that holds true when both sets are infinite dimensional. We adjust our definitions to accommodate infinite dimensions. These definitions reduce to the usual 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 that 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, 7, and 8. In Section 6 we solve a system of equations which require infinite dimensional splitting.