Solutions of nonlinear planar elliptic problems with triangle symmetry

[摘要] In this paper we continue the study of the nodal domain structure of doubly periodic solutions of certain nonlinear elliptic problems initiated in Fife et al. (Physica D 100 (1997), 257-278). More precisely, we consider small amplitude solutions of Delta u + lambda f(u) = 0 in 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 et al. (Physica D 100 (1997), 257-278) as well as other nodal domain structures. (C) 1997 Academic Press.

[发布日期] 1997-05-01