Solutions of Nonlinear Planar Elliptic Problems with Triangle Symmetry

In this paper we continue the study of the nodal domain structure of doubly periodic solutions of certain nonlinear elliptic problems initiated in Fifeet al.(Physica D100(1997), 257–278). More precisely, we consider small amplitude solutions ofΔu+λf(u)=0 in 2whose nodal domains consist of equilateral triangles tiling the plane. If this equation is suitably perturbed, then for genericfwe 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 Fifeet al.(Physica D100(1997), 257–278) as well as other nodal domain structures