TY - JOUR

T1 - The structure of Rabinowitz' global bifurcating continua for problems with weak nonlinearities

AU - Bari, Rehana

AU - Rynne, Bryan Patrick

PY - 1997/12

Y1 - 1997/12

N2 - Rabinowitz' global bifurcation theorem shows that for a large class of nonlinear eigenvalue problems a continuum (i.e., a closed, connected set) of solutions bifurcates from the trivial solution at each eigenvalue (or characteristic value) of odd multiplicity of the linearized problem (linearized at the trivial solution). Each continuum must either be unbounded, or must meet some other eigenvalue. This paper considers a class of such nonlinear eigenvalue problems having simple eigenvalues and a “weak” nonlinear term. A result regarding the location of the continua is obtained which shows, in particular, that in this case the bifurcating continua must be unbounded. Also, under further differentiability conditions it is shown that the continua are smooth, 1-dimensional curves and that there are no non-trivial solutions of the equation other than those lying on the bifurcating continua.

AB - Rabinowitz' global bifurcation theorem shows that for a large class of nonlinear eigenvalue problems a continuum (i.e., a closed, connected set) of solutions bifurcates from the trivial solution at each eigenvalue (or characteristic value) of odd multiplicity of the linearized problem (linearized at the trivial solution). Each continuum must either be unbounded, or must meet some other eigenvalue. This paper considers a class of such nonlinear eigenvalue problems having simple eigenvalues and a “weak” nonlinear term. A result regarding the location of the continua is obtained which shows, in particular, that in this case the bifurcating continua must be unbounded. Also, under further differentiability conditions it is shown that the continua are smooth, 1-dimensional curves and that there are no non-trivial solutions of the equation other than those lying on the bifurcating continua.

UR - http://www.scopus.com/inward/record.url?scp=0347706770&partnerID=8YFLogxK

U2 - 10.1112/S0025579300012717

DO - 10.1112/S0025579300012717

M3 - Article

SN - 0025-5793

VL - 44

SP - 419

EP - 433

JO - Mathematika

JF - Mathematika

IS - 2

ER -