Newton's Method and Morse Index for semilinear Elliptic PDEs

In this paper we primarily consider the family of elliptic PDEs ∆u+ f(u) = 0 on the square region Ω = (0, 1)× (0, 1) with zero Dirichlet boundary condition. Following our previous analysis and numerical approximations which relied on the variational characterization of solutions as critical points of an “action” functional, we consider Newton’s method on the gradient of that functional. We use a Galerkin expansion, in eigenfunctions of the Laplacian, to find solutions of arbitrary Morse index. Taking f ′(0) to be a bifurcation parameter, we analyze the bifurcations from the trivial solution, u ≡ 0, using symmetry arguments and our numerical algorithm. The Morse index of the approximated solutions is provided and support is found concerning several existence and nodal structure conjectures. We discuss the applicability of this method to find critical points of functionals in general.