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

We are concerned with the properties of weak solutions of the stationary Schr"odinger equation $-\mathrm{\Delta }u+Vu=f(u)$, $u\in H^1($\mathbb{R}$^N)\cap L^\infty($\mathbb{R}$^N)$,where$V$isH\ddot{o}ldercontinuousand$\inf V>0$. 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 $(0,\mathrm{\infty })$ 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.