Some variational methods for quasilinear elliptic differential equations on unbounded domains.

摘要

his dissertation is concerned with some variational methods from nonlinear functional analysis and their applications to elliptic Dirichlet boundary value problems on unbounded cylindrical domains using.;The loss of compactness of the Sobolev embedding theorems on unbounded domains renders variational techniques more delicate. In 1984 P. L. Lions introduced the local case of the concentration compactness principle with applications to elliptic problems on unbounded domains. Lions' method was to show weak continuity of the functionals on minimizing sequences, rather than on the whole space, proving exceptional properties of such sequences.;In 1989 and 1990, M. Schechter and K. Tintarev published a series of papers on eigenvalues of nonlinear functionals of the type ;This dissertation combines the two methods to obtain results for elliptic problems on unbounded cylinders with a nonhomogeneous nonlinearity. Similar results to those of Schechter and Tintarev are obtained under different hypotheses aimed at applications to problems on unbounded domains. A modification of the concentration compactness lemma is introduced permitting application of mountain pass lemma's to autonomous problems. Two applications are then considered. First we give sufficient conditions on f for the existence of a solution to the Dirichlet boundary value problem on