Abstract The equation Δ u + λ u + g ( λ , u ) u = 0 is considered in a bounded domain in R 2 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 λ 0 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 involving the corresponding eigenfunction u 0 . In the case when λ 0 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.