Nehari manifold optimization and its application for finding unstable solutions of semilinear elliptic PDEs
<|reference_start|>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 the generic 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 iteration points always lie on the Nehari manifold; the other is the tangential search direction to decrease the generic 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) overcoming 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 H\'enon equation and the stationary nonlinear Schr\"odinger equation. In particular, the symmetry-breaking phenomenon of the ground states of H\'enon equation is explored numerically in 1D and 2D with interesting numerical findings on the critical value of symmetry-breaking reported.<|reference_end|>
@article{chen2024nehari,
    title={Nehari manifold optimization and its application for finding unstable
  solutions of semilinear elliptic PDEs},
    author={Zhaoxing Chen, Wei Liu, Ziqing Xie, Wenfan Yi},
    journal={arXiv preprint arXiv:2404.09892},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.09892},
    primaryClass={math.NA cs.NA}
}