Semilinear equations in bounded cylinders: Morse index and bifurcation from one-dimensional solutions

Abstract: In this paper, we study semilinear elliptic equations in domains where there is a natural class of solutions, which depend only on one variable, and whose simple geometry reflects the geometry of the domain. We prove that under quite general assumptions, other types of solutions also exist. More precisely, we consider one-dimensional solutions in bounded cylinders and, combining a suitable separation of variables with the theory of ordinary differential equations, we show how to compute the Morse index of such solutions. The Morse index is then used to prove local and global bifurcation results.