ON SELF-SIMILAR SOLUTIONS OF SEMILINEAR HEAT EQUATIONS AND SEPARATION STRUCTURE FOR RELATED ELLIPTIC EQUATIONS

Abstract

We establish that if $n\geq3$ and $p\gt 1$ are large enough, then for each $\alpha\gt 0$ the elliptic equation $\Delta u+\frac12x\cdot\nabla u+\frac m2u+|x|^lu^p=0$ in $\bf R^n$ with $l\gt -2$ and $m=\frac{l+2}{p-1}$ possesses a positive radial solution $u_\alpha$ with $u_\alpha(0)=\alpha$ such that (i) $u_\beta\gt u_\alpha$ for $\beta\gt \alpha\gt 0$; (ii) for every $\alpha\gt 0$, $r^mu_\alpha(r)\rightarrow \ell$ as $r\rightarrow\infty$ for some $0\lt \ell=\ell(\alpha) \lt l =l(\alpha) \lt L$; (iii) $l(\alpha)$ is a one-to-one and onto increasing map from $(0, \infty)$ to $(0,L)$, where $L=[m(n-2-m)]^{1/(p-1)}$