A pointwise lower bound for positive solutions of a Schrodinger equation in R(N)

[摘要] Strong positivity of the bounded inverse (-A)(-1) of a Schrodinger operator -A = -Delta + q(x). in L(2)(R(N)) is proved in the following form: If -Au = f greater than or equal to 0 in L(2)(R(N)) with f not equivalent to 0, then u greater than or equal to c phi(1) a.e. in R(N). Here, phi(1) is the positive eigenfunction associated with the principal eigenvalue lambda(1) of -A, and c is a positive constant. It is shown that this result is valid if and only if the potential q(x), which is assumed to be strictly positive and locally bounded, has a sufficiently fast 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 eigen function phi(1). In particular, problems of existence and uniqueness are addressed. (C) 1997 Academic Press

[发布日期] 1997-01-20