Variational eigenvalues of quasi-linear operators and critical groups
Boundary value problems involving variational type quasilinear elliptic operators arise naturally in a wide range of applications. The solutions of such problems can often be obtained as the critical points of an associated variational functional defined on a suitable function space. In recent years infinite dimensional Morse theory, which is one of the most important tools in critical point theory, has been used to study semilinear problems with striking success, but unfortunately this is not so in the quasilinear case. Basic Morse theory describes the local behavior of a functional near an isolated critical point by a sequence of groups called the critical groups, and there are serious difficulties in determining the critical groups, both at finite critical points and at infinity, for functionals arising from quasilinear problems. Standard tools such as the generalized Morse lemma and the shifting theorem can no longer be applied since the domains of such functionals are generally not Hilbert spaces. Moreover, a complete description of the spectrum of a quasilinear operator is usually not available and there are no eigenspaces to work with. The goal of this proposal is to determine the critical groups in a broad class of variational form quasilinear problems, both resonant and nonresonant, and apply the results to obtain existence, multiplicity, and qualitative properties of solutions such as the number of nodal domains.
Nonlinear partial differential equations is one of the most applicable and highly interdisciplinary areas of mathematics. They are widely used to model fundamental processes in science and engineering. While tools for numerically computing solutions of quasilinear equations have been developed extensively, there is very little analytical theory currently available. Our goal is to establish parts of the needed theory and to investigate the structure of solutions. The knowledge gained by this project will have applications to a wide variety of physical systems.
