Dichotomy of stable radial solutions of $-\Delta u=f(u)$ outside a ball

Abstract

This paper is devoted to the study of stable radial solutions of $-\Delta u=f(u) \mbox{ in } \mathbb{R}^N\setminus B_1=\{ x\in \mathbb{R}^N : \vert x\vert\geq 1\}$, where $f\in C^1(\mathbb{R})$ and $N\geq 2$. We prove that such solutions are either large [in the sense that $\vert u(r)\vert \geq M r^{-N/2+\sqrt{N-1}+2}\ $, if $2\leq N\leq 9$; $\vert u(r)\vert \geq M \log (r)\ $, if $N=10$; $\vert u(r)-u_\infty \vert \geq M r^{\-N/2+\sqrt{N-1}+2}\ $, if $N\geq 11$; $\forall r\geq r_0$, for some $M>0$, $r_0\geq 1$] or small [in the sense that $\vert u(r)\vert \leq M\log (r)\ $, if $N=2$; $\vert u(r)-u_\infty \vert \leq M r^{\-N/2-\sqrt{N-1}+2}$;\, if $N\geq 3$; $\forall r\geq 2$, for some $M>0$], where $u_\infty=\lim_{r\rightarrow \infty}u(r)\in [-\infty,+\infty]$. These results can be applied to stable outside a compact set radial solutions of equations of the type $-\Delta u=g(u) \mbox{ in } \mathbb{R}^N$. We prove also the optimality of these results, by considering solutions of the form $u(r)=r^\alpha$ or $u(r)=\log (r)$, $\forall r\geq 1$, where $\alpha \in \mathbb{R} \setminus \{ 0\}$.