Nehari manifold optimization and its application for finding unstable solutions of semilinear elliptic PDEs

A Nehari manifold optimization method (NMOM) is introduced for finding 1-saddles, i.e., saddle points with the Morse index equal to one, of a generic nonlinear functional in Hilbert spaces. Actually, it is based on the variational characterization that 1-saddles of this functional are local minimizers of the same functional restricted on the associated Nehari manifold. The framework contains two important ingredients: one is the retraction mapping to make the iterative points always lie on the Nehari manifold; the other is the tangential search direction to decrease the functional with suitable step-size search rules. Particularly, the global convergence is rigorously established by virtue of some crucial analysis techniques (including a weak convergence method) that overcome difficulties in the infinite-dimensional setting. In practice, combining with an easy-to-implement Nehari retraction and the negative Riemannian gradient direction, the NMOM is successfully applied to compute the unstable ground-state solutions of a class of typical semilinear elliptic PDEs, such as the stationary nonlinear Schr\"odinger equation and the H\'enon equation. In particular, the symmetry-breaking phenomenon of the ground states of the H\'enon equation is explored numerically in 1D and 2D with interesting numerical findings on the critical value of the symmetry-breaking reported.