Solutions of semilinear elliptic equations with one isolated singularity

Abstract

If $f$ is either given by $(1+u)^p$ for some $\frac{N+2}{N-2}< p < \frac{N+1}{N-3}$, $N\geq 3$ or if $f$ is given by $e^u$ when $N=3$, we prove the existence of a positive weak solution of $ \Delta u + \lambda f(u) =0 $ which is defined in the unit ball of ${\Bbb R}^N$, has $0$ boundary data and has a nonremovable prescribed singularity at some point $x_0$ close to the origin.