An Extension of Maximum and Anti-Maximum Principles to a Schrodinger Equation in R~2

摘要

Strong maximum and anti-maximum principles are extended to weak L~2(R~2)-solutions u of the Schrodinger equation -#DELTA#u+q(x)u-#lambda#u = f(x) in L~2(R~2) in the following form: Let #phi#_1 denote the positive eigenfunction associated with the principal eigenvalue #lambda#_1 of the Schrodinger operator A =- #DELTA#+q(x) (centre dot) in L~2(R~2). Assume that q(x) (ident to) q(|x|), f is a "sufficiently smooth" perturbation of a radially symmetric function, f (not ident to) 0 and 0 <= f/#phi#_1 <= C (ident to) const a.e. in R~2. Then there exists a positive number #delta# (depending upon f) such that, for every #lambda# belong to (is member of) the set (-infinity,#lambda#_1 + #delta#) with #lambda# (not=) #lambda#_1, the inequality (#lambda#_1-#lambda#)u >= c#phi#_1 holds a.e. in R~2, where c is a positive constant depending upon f and #lambda#. 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|->infinity. 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 #phi#_1. In particular, problems of existence and uniqueness are addressed.