SELF-SIMILAR SOLUTIONS OF A SEMILINEAR HEAT EQUATION

Abstract

In this note we classify positive solutions of an equation$$\Delta u + \frac 1 2 x \cdot \nabla u + \frac 1 {p-1} u - |u|^{p-1} u =0\quad\text{in} \quad R^N,$$where $1\lt p\lt (N+2)/N$.

Under the assumption that $|x|^{2/(p-1)} u(x)$ is uniformlybounded in $ R^N$, we show that as $r=|x|$ tends to $\infty$, $r^{2/(p-1)} u(r \sigma)$converges uniformly to a continuous function $A(\sigma)$ on$S^{N-1}$. Conversely we also show that given any nonnegative continuous function$A(\sigma)$ on $S^{N-1}$,there exists a unique positive solution with that property.