Smooth bifurcation branches of solutions for a Signorini problem Original Research Article

We study a bifurcation problem for the equation Δu+λu+g(λ,u)u=0Δu+λu+g(λ,u)u=0 on a rectangle with Signorini boundary conditions on a part of one edge and mixed (zero Dirichlet and Neumann) boundary conditions on the rest of the boundary. Here λ∈Rλ∈R is the bifurcation parameter, and gg is a small perturbation. We prove, under certain assumptions concerning an eigenfunction u0u0 corresponding to an eigenvalue λ0λ0 of the linearized equation with the same nonlinear boundary conditions, the existence of a local smooth branch of nontrivial solutions bifurcating from the trivial solutions at λ0λ0 in the direction of u0u0. The contact sets of these nontrivial solutions are intervals which change smoothly along the branch. The main tool of the proof is a local equivalence of the unilateral BVP to a system consisting of a corresponding classical BVP and of two scalar equations. To this system classical Crandall–Rabinowitz type local bifurcation techniques (scaling and Implicit Function Theorem) are applied.