Non-energy semi-stable radial solutions

This paper is devoted to the study of semi-stable radial solutions $u\notin H^1(B_1)$ of $-\Delta u=f(u) \mbox{in} \overline{B_1}\setminus \{0\}=\{x\in \mathbb{R}^N : 0<\vert x\vert\leq 1\}$, where $f\in C^1(\mathbb{R})$ and $N\geq 2$. We establish sharp pointwise estimates for such solutions. In addition, we prove that in dimension $N=2$, any semi-stable radial weak solution of $-\Delta u=f(u)$, posed in $B_1$ with Dirichlet data $u|_{\partial B_1}=0$, is regular.