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

✍ Scribed by Bénédicte Alziary; Peter Takáč

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 734 KB
• Volume: 133
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1996.3205

✦ Synopsis

Strong positivity of the bounded inverse (&A) &1 of a Schro dinger operator &A=&2+q(x) v in L 2 (R N ) is proved in the following form: If &Au=f 0 in L 2 (R N ) with f 0, then u c. 1 a.e. in R N . Here, . 1 is the positive eigenfunction associated with the principal eigenvalue * 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| Ä . 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.