A Pointwise Lower Bound for Positive Solutions of a Schrödinger Equation in N

Strong positivity of the bounded inverse (− )−1of a Schrödinger operator − =−Δ+q(x) • inL2( N) is proved in the following form: If − u=f 0 inL2( N) withf 0, thenu c 1a.e. in N. Here, 1is the positive eigenfunction associated with the principal eigenvalueλ1of − , andcis a positive constant. It is shown that this result is valid if and only if the potentialq(x), which is assumed to be strictly positive and locally bounded, has a sufficiently fast growth as x→∞. This result is applied to linear and nonlinear elliptic boundary value problems in strongly ordered Banach spaces, whose positive cone is generated by the eigenfunction 1. In particular, problems of existence and uniqueness are addressed.