Symmetry-Breaking for Positive Solutions of Semilinear Elliptic Equations

In a recent interesting paper, GIDAS, NI, and NIRENBERG [2] proved that positive solutions of the Dirichlet problem for second-order semi-linear elliptic equations on balls must themselves be spherically symmetric functions. Here we consider the bifurcation problem for such solutions. Specifically, we investigate the ways in which the symmetric solution can bifurcate into a nonsymmetric solution; when this happens, we say that the symmetry "breaks." To carry out this program, we rely on certain results in [6], where we studied the kernel of the associated linearized operator. This enables us to give some necessary conditions for symmetry to break. We also find a class of functionsf where these conditions are also sufficient; see equation (18) below. The problem is, of course, to show that, when zero comes into the spectrum of the linearized operator and our conditions are fulfilled, bifurcation into the non-radial direction actually occurs. This is done by showing first that the "bifurcation curve" is a smooth manifold near the bifurcation point, and then appealing to a theorem of VANDERBAUWHEDE [8], which gives sufficient conditions for bifurcation to occur in the presence of symmetries.