Computations of critical groups and applications to asymptotically linear wave equation and beam equation
The use of critical groups to study the existence and multiplicity of solutions to partial differential equations arising from the application of a variational principle relies heavily on Morse theory. However, there is a problem in the case of strongly indefinite functionals. A new theory is developed for the Morse index for strongly indefinite functionals. This theoretical development constitutes some half of the paper. Two problems, the asymptotically linear wave equation and the asymptotically linear beam equation, to which class of equations the theoretical development is directed, are given. For the former nine cases for which at least one nontrivial weak solution exists are identified. With an additional consumption, the existence of a nonzero nondegenerate periodic solution, a further nonzero solution occurs. Similar results are found for the latter which is observed to be very similar to the former under the conditions stated.
scientific article; zbMATH DE number 2137692
scientific article