An Extension of Maximum and Anti-Maximum Principles to a Schrödinger Equation in 2

Strong maximum and anti-maximum principles are extended to weak L2 ( 2)-solutions u of the Schrödinger equation −Δu+q(x) u−λu=f(x) in L2( 2) in the following form: Let 1 denote the positive eigenfunction associated with the principal eigenvalue λ1 of the Schrödinger operator =−Δ+q(x) • in L2( 2). Assume that q(x)≡q(x), f is a “sufficiently smooth” perturbation of a radially symmetric function, f 0 and 0 f/ 1 C≡const a.e. in 2. Then there exists a positive number δ (depending upon f) such that, for every λ (−∞, λ1+δ) with λ≠λ1, the inequality (λ1−λ) u c 1 holds a.e. in 2, where c is a positive constant depending upon f and λ. It is shown that such an inequality is valid if and only if the potential q(x), which is assumed to be strictly positive and locally bounded, has a superquadratic 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.