Nonradial Solutions of a Semilinear Elliptic Equation in Two Dimensions

Abstract

: We establish existence of an infinite family of exponentially-decaying non-radial $C^2$ solutions to the equation $\Delta u + f(u) = 0$ on $R^2$ for a large class of nonlinearities $f$. These solutions have the form $u(r,\theta )=e^{i m\theta }w(r)$, where $r$ and $\theta$ are polar coordinates, $m$ is an integer, and $w:[0,\infty ) \to R$ is exponentially decreasing far from the origin. We prove there is a solution with each prescribed number of nodes.