Low-dimensional qualitative and numerical approximation of bifurcation in a semilinear elliptic problem

Abstract A nonlinear boundary value problem, exhibiting a nonlocal bifurcation in its solution set, is considered. It is shown that for some values of the problem parameters there is only the zero solution, while for other values of these parameters two nontrivial solutions also appear. These nontrivial solutions do not lie in small neighborhoods of the zero function, so that the bifurcation analysis is not local to the origin. The method of reduction to (alternative) finite-dimensional problems is applied to give a numerical description of the bifurcation, and in particular, to describe a surface of bifurcating solutions. In addition the solution set of the 1-dimensional Galerkin approximation is also shown to have the same qualitative nature as the solution set of the full problem.