An eigenvalue problem for nonlinear elliptic partial differential equations
Melvyn S. Berger · Transactions of the American Mathematical Society · 1965
(1) U JOGD= Du JOG= = Dm'UIaG = 0, where A is a real number and f(t, x) is a real-valued function defined on R' x G with f (0, x) 0. Iff (u, x) u, the study of the boundary value problem (1) forms the foundation of the spectral analysis of A, a problem of great importance both in mathematics and its applications. If f(u, x) does not depend on u in a linear manner, one enters the relatively uncharted world of nonlinear functional analysis. We shall be concerned with the existence of real-valued nontrivial solutions of (1), i.e. eigenfunctions. There are basically two different approaches to such nonlinear existence problems: first the methods of fixed point theory and other topological principles used with success in the study of elliptic partial differential equations since the pioneering work of S. Bernstein and J. Schauder; second the variational method, dating back to Gauss, Dirichlet and Riemann, and currently, in combination with the new methods of Sobolev spaces, undergoing a rapid development. Throughout this study we shall rely on this latter approach. For second order operators A, one of the first treatments of boundary value problems of the type considered here was given by A. Hammerstein [17], in 1930, as an application of his study of nonlinear integral equations. The approach used in this dissertation is based on a direct study of elliptic differential operators without recourse to integral equations and Green's functions. By focusing attention on the so-called generalized solutions of (1), we are able to use a variety of Hilbert spaces in our study and to eliminate the auxiliary analytic machinery of a priori estimates, and smoothness properties on the domain G. The following questions will occupy our attention in this study. (i) (Existence Problem). Under what restrictions on the function f(t, x) does