Some existence theorems for nonlinear eigenvalue problems associated with elliptic equations

Abstract

The existence of harmonic functions in a region D of the plane satisfying nonlinear Neumann boundary conditions of the form Ou]On=f(u) where n is the outwardly directed unit normal to the boundary OD of D has been studied by T. CAgLrMAN ([4]), NAKAMORI & SOYAMA ([16]), K. KLINGELH/~FER ([9], [10], [11]), and, in the case when D is the unit disk andf i s of a highly specialized form which also depends on the harmonic conjugate of u, by LEVI-CIVITA ([13]) in his study of periodic progressing water waves. CARLEMAN and NAKAMOm & SUYAMA considered the ease when f ' (u)<0 and found existence results for singular solutions. KLINGELIq61~Im, using the contraction mapping principle, developed existence and uniqueness theorems for regular solutions under assumptions on f (u) which, roughly speaking, require that i f (u) stay away from the (necessarily positive) eigenvalues of the linear problem (the Steklov problem [2], [19]) Ou]On=2u; that is, ;tk < f ' (u) < 2k+ 1 where 2k, 2k+ 1 are consecutive Steldov eigenvalues. These results, however, never yield nontrivial solutions to the problem in the case that f(0) = 0; LEvI-CIVlTA, on the other hand, found (in his special problem) nontrivial solutions even though the problem had the trivial solution u= 0. The problem of finding nontrivial solutions to elliptic equations under such nonlinear boundary conditions (actually a problem in bifurcation theory) is the subject of this paper. Various criteria for uniqueness have been found by many authors including MARTIN, LEVlN, DUNNINGER, and CUSHING (see [5] for bibliography). The problem is the following: to satisfy the equation