Some semilinear elliptic equations with nonlinear boundary conditions: Radial solutions and simple symmetry-breaking
Peter Grindrod · IMA Journal of Applied Mathematics · 1995
The subject of nonnegative solutions for semilinear equations has received 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 defined on spherically symmetric domains are presented. The author considers the existence of radially symmetric solutions as a function of domain size, and also shows that infinitesimal symmetry-breaking bifurcations 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 division for example), this result suggests attention should be further focused upon modelling nonlinear boundary conditions.