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 $-\Delta u=f(u)$,posedin$B_1$withDirichletdata$u|_{\partial B_1}=0$, is regular.