Smooth bifurcation for a Signorini problem on a rectangle

Abstract

We study a parameter depending semilinear elliptic PDE 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. We describe smooth branches of smooth nontrivial solutions bifurcating from the trivial solution branch in eigenvalues of the linearized problem. In particular, the contact sets of these nontrivial solutions are intervals which change smoothly along the branch. The main tools of the proof are first a certain local equivalence of the unilateral BVP to a system consisting of a corresponding classical BVP and of two scalar equations (which determine the ends of the contact intervals), and secondly an application of the classical Crandall-Rabinowitz type local bifurcation techniques (scaling and application of the Implicit Function Theorem) to that system.