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We are concerned with the properties of weak solutions of the stationarySchr\"odinger equation $-\Delta u + Vu = f(u)$, $u\in H^1(\mathbb{R}^N)\capL^\infty(\mathbb{R}^N)$, where $V$ is H\"older continuous and $\inf V>0$.Assuming $f$ to be continuous and bounded near $0$ by a power function withexponent larger than $1$ we provide precise decay estimates at infinity forsolutions in terms of Green's function of the Schr\"odinger operator. In somecases this improves known theorems on the decay of solutions. If $f$ is alsoreal analytic on $(0,\infty)$ we obtain that the set of positive solutions islocally path connected. For a periodic potential $V$ this implies that thestandard variational functional has discrete critical values in the low energyrange and that a compact isolated set of positive solutions exists, underadditional assumptions.

Precise exponential decay for solutions of semilinear elliptic equations  and its effect on the structure of the solution set for a real analytic  nonlinearity