An Extension of Maximum and Anti-Maximum Principles to a Schrödinger Equation in R2

✍ Scribed by Bénédicte Alziary; Jacqueline Fleckinger-Pellé; Peter Takáč

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 228 KB
• Volume: 156
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1998.3609

✦ Synopsis

Strong maximum and anti-maximum principles are extended to weak L 2 (R 2 )solutions u of the Schro dinger equation &2u+q(x) u&*u= f (x) in L 2 (R 2 ) in the following form: Let . 1 denote the positive eigenfunction associated with the principal eigenvalue * 1 of the Schro dinger operator A=&2+q(x) v in L 2 (R 2 ). Assume that q(x)#q(|x|), f is a ``sufficiently smooth'' perturbation of a radially symmetric function, f 0 and 0 fÂ. 1 C#const a.e. in R 2 . Then there exists a positive number $ (depending upon f ) such that, for every * # (& , * 1 +$) with *{* 1 , the inequality (* 1 &*) u c. 1 holds a.e. in R 2 , where c is a positive constant depending upon f and *. 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| 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.