Elliptic Equation with a Singular Potential in a Domain with a Conic Point

被引：0

作者：

Khudaikuliev, B. A. ^{[1
]}

机构：

[1] Turkmen State Univ, Ashkhabad, Turkmenistan

来源：

MATHEMATICAL NOTES | 2012年 / 92卷 / 5-6期

关键词：

elliptic equation; singular potential; conic domain; conic point; Laplace operator; Beltrami operator; Dirichlet boundary condition; Cauchy's inequality; Holder's inequality;

D O I：

10.1134/S0001434612110272

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

This paper deals with the behavior of the nonnegative solutions of the problem -Delta u = V (x)u, u vertical bar(partial derivative Omega) = phi(x) in a conical domain Omega subset of R-n, n >= 3, where 0 <= V (x) is an element of L-1(Omega), 0 <= phi(x) is an element of L-1(partial derivative Omega) and phi(x) is continuous on the boundary partial derivative Omega. It is proved that there exists a constant C-star(n) = (n - 2)(2)/4 such that if V-0(x) = (c + lambda(1))vertical bar x vertical bar(-2), then, for 0 <= c <= C-star(n) and V (x) <= V-0(x) in the domain Omega, this problem has a nonnegative solution for any nonnegative boundary function phi(x) is an element of L-1(partial derivative Omega); for c > C-star(n) and V (x) >= V-0(x) in Omega, this problem has no nonnegative solutions if phi(x) > 0. DOI: 10.1134/S0001434612110272

引用

页码：820 / 829

页数：10