Nonradial Solutions of a Semilinear Elliptic Equation in Two Dimensions

We establish existence of an infinite family of exponentially-decaying non-radial C2 solutions to the equation Δu+ƒ(u)=0 on 2 for a large class of nonlinearities ƒ. These solutions have the form u(r, θ)=eimθw(r), where r and θ are polar coordinates, m is an integer, and w: [0, ∞)→ is exponentially decreasing far from the origin. We prove there is a solution with each prescribed number of nodes.