An Extension of Maximum and Anti-Maximum Principles to a Schrödinger Equation in R2
An Extension of Maximum and Anti-Maximum Principles to a Schrödinger Equation in R2
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R2 中薛定谔方程的极大值和反极大值原理的扩展
DOI：
10.1006/jdeq.1998.3609
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发表时间：
1999
影响因子：
2.4
通讯作者：
P. Takáč
中科院分区：
文献类型：
--
作者：
B. Alziary;J. Fleckinger;P. Takáč
将强极大和反极大原理推广到L2（R2）中薛定谔方程-Δu+q（x）u-λu=f（x）的弱L2（R2）-解u的情形，其形式如下：设λ 1表示L2（R2）中与薛定谔算子A =-Δ+q（x）·的主本征值λ1有关的正本征函数。设q（x）= q（|X|），f是径向对称函数的“足够光滑”扰动，f 0且0 f/1 C≡const a.e.在R2中。则存在一个正数δ（取决于f），使得对于每个λ∈（−∞，λ1+δ）且λ <$λ1，不等式（λ1−λ）u <$c <$1成立a.e.在R2中，其中c是取决于f和λ的正常数。证明了这样一个不等式成立的充要条件是势函数q（x）是严格正的且局部有界的，且具有超二次增长，|X| →∞。这个结果被应用到强序Banach空间中的线性和非线性椭圆边值问题，其正锥由特征函数φ1生成。特别是，存在性和唯一性的问题得到解决。
Abstract Strong maximum and anti-maximum principles are extended to weak L2 ( R 2)-solutions u of the Schrodinger equation −Δu+q(x) u−λu=f(x) in L2( R 2) in the following form: Let ϕ1 denote the positive eigenfunction associated with the principal eigenvalue λ1 of the Schrodinger operator A =−Δ+q(x) • in L2( R 2). Assume that q(x)≡q(|x|), f is a “sufficiently smooth” perturbation of a radially symmetric function, f≢0 and 0⩽f/ϕ1⩽C≡const a.e. in R 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⩾cϕ1 holds a.e. in R 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 ϕ1. In particular, problems of existence and uniqueness are addressed.