tailieunhanh - Báo cáo hóa học: " Uniqueness of positive solutions to a class of semilinear elliptic equations"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí sinh học đề tài : Uniqueness of positive solutions to a class of semilinear elliptic equations | Li and Zhou Boundary Value Problems 2011 2011 38 http content 2011 1 38 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access Uniqueness of positive solutions to a class of semilinear elliptic equations Chunming Li and Yong Zhou Correspondence yzhoumath@ Department of Mathematics Zhejiang Normal University Jinhua 321004 Zhejiang PR China Springer Abstract In this article we consider the uniqueness of positive radial solutions to the Dirichlet boundary value problem Au f x u g x x Vu 0 x e o u 0 x e i íỉ where Q denotes an annulus in Rn n 3 . The uniqueness criterion is established by applying shooting method. Keywords positive solution semilinear elliptic equation uniqueness 1 Introduction This article is concerned with the positive radial solutions to a class of semilinear elliptic equations Au f x u g x x Vu 0 x e u 0 x e dtt 1 1 where ít x x e R a x b a and b are positive real numbers f e C1 0 X 0 and g 0 R is differentiable. Equation describes stationary states for many reaction-diffusion equations. The absence of positive solutions to the elliptic equations also means that the existing solutions oscillate which is also important information in applications. In recent years there is a widespread concern over the positive solutions to the Dirichlet boundary value problem when g x 0 . Au If x u 0 u 0in u 0 x e 3ft. 1 2 When the nonlinear term just depends on u the uniqueness of has been exhaustively studied see 1-6 . In 1985 the uniqueness of was discussed in different domains by Ni and Nussbaum 7 to the case whenf depends on x and u f x u 0 andf x u satisfies some growth conditions. Erbe and Tang 8 presented a new uniqueness criterion using a shooting method and Sturm comparison theorem. So far it seems that nobody considers the uniqueness to problem . Inspired by the above articles the aim of the present article is to establish some simple criteria for the uniqueness of positive .