  We consider the nonlinear Schr\"{o}dinger equation $-\Delta u + V(x) u =
\Gamma(x) |u|^{p-1}u$ in $\R^n$ where the spectrum of $-\Delta+V(x)$ is
positive. In the case $n\geq 3$ we use variational methods to prove that for
all $p\in (\frac{n}{n-2},\frac{n}{n-2}+\eps)$ there exist distributional
solutions with a point singularity at the origin provided $\eps>0$ is
sufficiently small and $V,\Gamma$ are bounded on $\R^n\setminus B_1(0)$ and
satisfy suitable H\"{o}lder-type conditions at the origin. In the case $n=1,2$
or $n\geq 3,1<p<\frac{n}{n-2}$, however, we show that every distributional
solution of the more general equation $-\Delta u + V(x) u = g(x,u)$ is a
bounded strong solution if $V$ is bounded and $g$ satisfies certain growth
conditions.
