Nonradial Solutions of a Semilinear Elliptic Equation in Two Dimensions

✍ Scribed by J. Iaia; H. Warchall

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 668 KB
• Volume: 119
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1995.1101

✦ Synopsis

We establish existence of an infinite family of exponentially-decaying non-radial (C^{2}) solutions to the equation (\Delta u+f(u)=0) on (\mathbb{R}^{2}) for a large class of nonlinearities (f). These solutions have the form (u(r, \theta)=e^{\text {imit }} u(r)), where (r) and (\theta) are polar coordinates, (m) is an integer, and (w:[0, \infty) \rightarrow \mathbb{R}) is exponentially decreasing far from the origin. We prove there is a solution with each prescribed number of nodes. r 1995 Academic Press. inc.