Smooth bifurcation for variational inequalities based on Lagrange multipliers. (English) Zbl 1212.35174

Eisner, Jan; Kučera, Milan; Recke, Lutz

Differ. Integral Equ. 19, No. 9, 981-1000 (2006).

Summary: We prove a bifurcation theorem of Crandall-Rabinowitz type (local bifurcation of smooth families of nontrivial solutions) for general variational inequalities on possibly non-convex sets with infinite-dimensional bifurcation parameter. The result is based on local equivalence of the inequality to a smooth equation with Lagrange multipliers, on scaling techniques and on an application of the implicit function theorem. As an example, we consider a semilinear elliptic PDE with non-convex unilateral integral conditions on the boundary of the domain.

MSC:

35J87	Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators
35B32	Bifurcations in context of PDEs
47J15	Abstract bifurcation theory involving nonlinear operators
49J40	Variational inequalities

Keywords:

local bifurcation; smooth family of nontrivial solutions; general variational inequality; Lagrange multipliers