  We are concerned with the properties of weak solutions of the stationary
Schr\"odinger equation $-\Delta u + Vu = f(u)$, $u\in H^1(\mathbb{R}^N)\cap
L^\infty(\mathbb{R}^N)$, where $V$ is H\"older continuous and $\inf V>0$.
Assuming $f$ to be continuous and bounded near $0$ by a power function with
exponent larger than $1$ we provide precise decay estimates at infinity for
solutions in terms of Green's function of the Schr\"odinger operator. In some
cases this improves known theorems on the decay of solutions. If $f$ is also
real analytic on $(0,\infty)$ we obtain that the set of positive solutions is
locally path connected. For a periodic potential $V$ this implies that the
standard variational functional has discrete critical values in the low energy
range and that a compact isolated set of positive solutions exists, under
additional assumptions.
