SOME SEMILINEAR ELLIPTIC-EQUATIONS WITH NONLINEAR BOUNDARY-CONDITIONS- RADIAL SOLUTIONS AND SIMPLE SYMMETRY-BREAKING

The subject of nonnegative solutions for semilinear equations has rece ived considerable attention in recent years. A significant omission is the consideration of nonlinear boundary conditions, and their impact upon the structure of solutions. In this paper some such problems defi ned on spherically symmetric domains are presented. The author conside rs the existence of radially symmetric solutions as a function of doma in size, and also shows that infinitesimal symmetry-breaking bifurcati ons may occur in the simplest eigenmode. This is precluded in the case of homogeneous Neumann or Dirichlet problems for similar source terms . Given the importance of robust simple symmetry-breaking (in cell div ision for example), this result suggests attention should be further f ocused upon modelling nonlinear boundary conditions.