Direction and stability of bifurcating solutions for a Signorini problem

作者：Eisner Jan^*; Kucera Milan; Recke Lutz

来源：Nonlinear Analysis-Theory Methods & Applications, 2015, 113: 357-371.

DOI：10.1016/j.na.2014.09.032

• Signorini problem
• Variational inequality
• Bifurcation direction
• Stability of bifurcating solutions
• Exchange of stability

The equation Delta u+lambda u+ g(lambda, 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(lambda, 0) = 0 for lambda is an element of R, lambda 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 lambda(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 lambda(0) is the first eigenvalue and the branch goes to the right, we show that the bifurcating solutions are asymptotically stable in W-1,W-2-norm. The stability of the trivial solution is also studied and an exchange of stability is obtained.

• 出版日期2015-1