On equilibrium solutions of diffusion equations with nonlinear boundary conditions

The existence or the nonexistence of nonconstant stable equilibrium solutions for a diffusion equation with nonlinear Neumann boundary conditions is studied. We prove the nonexistence of nonconstant stable equilibria when the nonlinearity has a small Lipschitz constant or a second derivative of constant sign or either when the domain is a ball. We construct an example of existence for a connected domain with several disconnected boundary components.