Variational Equations

Abstract

This chapter deals with existence and comparison results for weak solutions of nonlinear elliptic and parabolic problems. The ideas and methods developed here will also be useful in the treatment of nonsmooth variational problems in later chapters. Section 3.1 deals with semilinear elliptic Dirichlet boundary value problems and may be considered as a preparatory section for Sect. 3.2 and Sect. 3.3, where general quasilinear elliptic and parabolic problems are treated. The purpose of Sect. 3.1 is to emphasize the basic ideas and to present various approaches without overburdening the presentation with too many technicalities. As an application of the general results of Sect. 3.2 combined with critical point theory, the existence of multiple and sign-changing solutions is proved in Sect. 3.4. Finally, in Sect. 3.5, the concept of sub-supersolutions is extended to some nonstandard elliptic boundary value problem, which in the one-space dimensional and semilinear case reduces to a second-order ordinary differential equation subject to periodic boundary conditions. The chapter concludes with bibliographical notes and further applications and extensions of the theory developed in the preceeding sections.