LOCAL ESTIMATES AND THE EXISTENCE OF MULTIPLE SOLUTIONS TO NONLINEAR ELLIPTIC PROBLEMS

Abstract

This chapter focuses on local estimates and the existence of multiple solutions to nonlinear elliptic problems. It reviews the existence of multiple solutions of boundary value problems for nonlinear elliptic partial differential equations of the given form. The chapter describes the way in which a method developed by L. Cesari can be used to study the existence of solutions to problem by considering the local behavior of a function. It discusses the way in which this approach leads to criteria for the existence of multiple solutions to problem. The chapter also discusses the evolution of the use of the alternative method from the existence of solutions to boundary value problems for nonlinear ordinary differential equations to the existence of multiple solutions for boundary value problems for nonlinear elliptic equations whose nonlinearity is a function of a partial derivative of u.