BIFURCATION AND SYMMETRY-BREAKING IN A NONLINEAR ELLIPTIC EQUATION IN A CYLINDER

We study a semi-linear elliptic equation in a cylindrical domain with mixed boundary conditions. We show that there exists a critical value of the length of the cylinder for which non-uniform solutions (with respect to the direction of the axis of the cylinder) appear. Moreover, these solutions are monotonic in this direction. We prove afterwards the existence of multiples solutions as the length of the cylinder increases. Some other qualitative properties and a result with supercritical exponent are also given.