Smooth bifurcation for variational inequalities based on the implicit function theorem

Abstract

We present a certain analog for variational inequalities of the classical result on bifurcation from simple eigenvalues of Crandall and Rabinowitz. In other words, we describe the existence and local uniqueness of smooth families of nontrivial solutions to variational inequalities, bifurcating from a trivial solution family at certain points which could be called simple eigenvalues of the homogenized variational inequality. If the bifurcation parameter is one-dimensional, the main difference between the case of equations and the case of variational inequalities (when the cone is not a linear subspace) is the following: For equations two smooth half-branches bifurcate, for inequalities only one. The proofs are based on scaling techniques and on the implicit function theorem. The abstract results are applied to a fourth order ODE with pointwise unilateral conditions (an obstacle problem for a beam with the compression force as the bifurcation parameter).