SOLUTIONS OF NONLINEAR PLANAR ELLIPTIC PROBLEMS WITH TRIANGLE SYMMETRY

In this paper we continue the study of the nodal domain structure of d oubly periodic solutions of certain nonlinear elliptic problems initia ted 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 plan e. If this equation is suitably perturbed, then for generic f we prove the existence of unique nearby solutions with triangle symmetry and s how how their nodal domain geometry breaks up. Furthermore, we treat t he non-generic rectangular cases which had to be excluded in Fife et a l. (Physica D 100 (1997), 257-278) as well as other nodal domain struc tures. (C) 1997 Academic Press.