Direction and stability of bifurcating solutions for a Signorini problem

The equation ∆u + λu + g(λ, u)u = 0 is considered in a bounded domain in R2 with a Signorini condition on a straight part of the boundary and with mixed boundary conditions on the rest of the boundary. It is assumed that g(λ, 0) = 0 for λ ∈ R, λ is a bifurcation parameter. A given eigenvalue of the linearized equation with the same boundary conditions is considered. A smooth local bifurcation branch of non-trivial solutions emanating at λ0 from trivial solutions is studied. We show that to know a direction of the bifurcating branch it is sufficient to determine the sign of a simple expression envolving the corresponding eigenfunction u0. In the case when λ0 is the first eigenvalue and the branch goes to the right, we show that the bifurcating solutions are asymptotically stable in W 1,2-norm. The stability of the trivial solution is also studied and an exchange of stability is obtained.