Solutions of nonlinear planar elliptic problems with triangle symmetry

9. Stanislaus Maier-Paape, Thomas Wanner:
   Solutions of nonlinear planar elliptic problems with triangle symmetry
   Journal of Differential Equations 136(1), pp. 1-34, 1997.

Abstract

In this paper we continue the study of the nodal domain structure of doubly periodic solutions of certain nonlinear elliptic problems initiated in Fife, Kielhöfer, Maier-Paape, Wanner (1997). More precisely, we consider small amplitude solutions of $\Delta u + \lambda f(u) = 0$ in $\mathbb{R}^2$ whose nodal domains consist of equilateral triangles tiling the plane. If this equation is suitably perturbed, then for generic $f$ we prove the existence of unique nearby solutions with triangle symmetry and show how their nodal domain geometry breaks up. Furthermore, we treat the non-generic rectangular cases which had to be excluded in Fife, Kielhöfer, Maier-Paape, Wanner (1997), as well as other nodal domain structures.

Links

The published version of the paper can be found at https://doi.org/10.1006/jdeq.1996.3240.

Bibtex

@article{maier:wanner:97a,
   author = {Stanislaus Maier-Paape and Thomas Wanner},
   title = {Solutions of nonlinear planar elliptic problems with
            triangle symmetry},
   journal = {Journal of Differential Equations},
   year = 1997,
   volume = 136,
   number = 1,
   pages = {1--34},
   doi = {10.1006/jdeq.1996.3240}
   }