PRECISE EXPONENTIAL DECAY FOR SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS AND ITS EFFECT ON THE STRUCTURE OF THE SOLUTION SET FOR A REAL ANALYTIC NONLINEARITY

摘要

We are concerned with the properties of weak solutions of the stationary Schrodinger equation -Delta u + Vu = f (u), u is an element of H-1 (R-N) boolean AND L-infinity (R-N), where V is Holder 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 Schrodinger operator. In some cases this improves known theorems on the decay of solutions. If f is also real analytic on (0, infinity), 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.