Pure Mathematics
Vol.4 No.05(2014), Article ID:14113,6 pages
DOI:10.12677/PM.2014.45029

Existence of Three Positive Solutions for a Class of Nonlinear Elliptic Systems

Gongming Wei, Yutong Chen, Xingli Zhang

Department of Mathematics, College of Science, University of Shanghai for Science and Technology, Shanghai

Email: gmweixy@163.com

Received: Jul. 6th, 2014; revised: Aug. 4th, 2014; accepted: Aug. 13th, 2014

ABSTRACT

Motivated by existence of solutions of single equation, in this paper we study the existence of multiple solutions of a class of nonlienar elliptic systems with nonhomogeneous boundary conditions. Using Guo-Krasnoselski’s fixed point theorem on cones, we prove that there exist at least three positive solutions for this class of nonlinear elliptic systems.

Keywords:Nonlinear Elliptic System, Positive Radial Solution, Fixed Point Theorem on Cones

Email: gmweixy@163.com

1. 引言

(1.1)

(H1)是关于的连续递增函数，即：当时，有成立；当时，有成立。

(H2) 对任意的，有成立。

(H3) 对任意的，有成立；对任意的，有成立。

(H4) 对任意的，有成立；对任意的，有成立。

(H5) 存在非负函数满足

2. 预备知识

(2.1)

，使，则(2.1)可化为

(2.2)

(2.3)

(1) 如果，则，且如果，则

(2) 如果，则，且如果，则

3. 主要结果的证明

, (3.1)

. (3.2)

(3.3)

，其中。由(H1)可知，当时，。又由(3.3)可得

(3.4)

(1) 存在一个子序列，其中的子序列，的子序列，并满足，对任意的，有。由(3.4)和(H5)可得

(2) 对任意的时，有(3.4)和假设(H5)可得

(3.5)

，即证明了对任意的，有下式成立

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