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 $-\Delta u=f(u) \text{ in } \mathbb {R}^N{\setminus } B_1=\{ x\in \mathbb {R}^N : \vert x\vert \ge 1\}$, where $f\in C^1(\mathbb {R})$ and $N\ge 2$. We prove that such solutions are either large [in the sense that $\vert u(r)\vert \ge M r^{-N/2+\sqrt{N-1}+2}\$, if $2\le N\le 9$; $\vert u(r)\vert \ge M \log (r)\$, if $N=10$; $\vert u(r)-u_\infty \vert \ge M r^{-N/2+\sqrt{N-1}+2}\$, if $N\ge 11$; $\forall r\ge r_0$, for some $M>0$, $r_0\ge 1$] or small [in the sense that $\vert u(r)\vert \le M\log (r)\$, if $N=2$; $\vert u(r)-u_\infty \vert \le M r^{-N/2-\sqrt{N-1}+2}$;  if $N\ge 3$; $\forall r\ge 2$, 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 $-\Delta u=g(u) \text{ in } \mathbb {R}^N$. We prove also the optimality of these results, by considering solutions of the form $u(r)=r^\alpha$ or $u(r)=\log (r)$, $\forall r\ge 1$, where $\alpha \in \mathbb {R} {\setminus } \{ 0\}$.