﻿ 具有Hardy奇异项的近共振Kirchhoff方程正解的多重性 Multiplicity of Positive Solutions for Kirchhoff Type of Equation with Hardy Singular Item

Pure Mathematics
Vol.05 No.02(2015), Article ID:14895,6 pages
10.12677/PM.2015.52004

Multiplicity of Positive Solutions for Kirchhoff Type of Equation with Hardy Singular Item

Hong Rong, Chunyu Lei, Hongmin Suo*

School of Science, Guizhou Minzu University, Guiyang Guizhou

Received: Feb. 13th, 2015; accepted: Feb. 25th, 2015; published: Mar. 2nd, 2015

ABSTRACT

In this paper, using the local minimum theorem and mountain pass lemma of variational methods, we study the Kirchhoff equation with Hardy singular term, and obtain multi- plicity results of solutions for this equation near resonance with principal eigenvalue.

Keywords:Kirchhoff Type of Equation, Hardy Singular Term, Variational Methods, Near Resonance

1. 引言

(1)

Ma et al. [1] 运用变分法得到了方程的正解。Zou [2] 使用局部极小原理和喷泉定理得到了方程非平凡解的存在性和多重性。特别地，Chen [3] 考虑下面的Kirchhoff方程

(2)

(3)

(4)

2. 预备知识

(5)

。 (6)

(7)

(8)

(9)

(10)

1)如果

2) 存在，当时，

1) 由引理2.1，可知引理2.2的条件(1)成立，证毕。

2) 对任意的，有

。因此，存在使得时，有，证毕。

(11)

(13)

(14)

(15)

3. 主要结果及其证明

(16)

(17)

(18)

(19)

(20)

(20)式表明。再由的连续性，可得是(1)的一个解，即

(21)

Multiplicity of Positive Solutions for Kirchhoff Type of Equation with Hardy Singular Item. 理论数学,02,21-27. doi: 10.12677/PM.2015.52004

1. 1. Ma, T.F. and Munoz Rivera, J.E. (2003) Positive solutions for a nonlinear elliptic transmission problem. Applied Ma-thematics Letters, 16, 243-248.

2. 2. He, X. and Zou, W. (2010) Multiplicity of solutions for a class of Kirchhoff type problems. Acta Mathematicae Appllicatae Sinica, 26, 387-394.

3. 3. Chen, J.Q. (2014) Multiple positive solutions to a class of Kirchhoff equation on with indefinite Nonlinearity. Nonlinear Analysis, 96, 134-145.

4. 4. Aubin, T. and Ekeland, I. (1984) Applied nonlinear analysis. Wiley, New York.

5. 5. Ambrosetti, A. and Rabinowitz, P.H. (1973) Dual variational methods in critical point theory and applications. Journal of Functional Analysis, 14, 349-381.