具有变号非线性项的多点边值问题的非平凡解

数学物理学报 ›› 2010, Vol. 30 ›› Issue (6): 1503-1513.

具有变号非线性项的多点边值问题的非平凡解

¹栾世霞|²万菲菲|³张新光

1.曲阜师范大学数学科学学院山东曲阜 273165|2.烟台南山学院理学院山东烟台 265713|3.烟台大学数学与信息科学学院山东烟台 264005

收稿日期:2008-12-08 修回日期:2009-12-14 出版日期:2010-12-25 发布日期:2010-12-25
基金资助:
国家自然科学基金(10771117)和山东省青年自然科学基金(Q2007A02, ZR2009AL016)资助

Nontrivial Solutions of Multi-point Boundary Value Problems with Sign-changing Nonlinear Terms

¹LUAN Shi-Xia, ²WAN Fei-Fei, ³ZHANG Xin-Guang

1.School of Mathematical Sciences, Qufu Normal University, Shandong Qufu 273165|2.College of Science, Yantai Nanshan College, Shandong Yantai 265713|3.Department of Mathematics and Informational Sciences, Yantai University, Shandong Yantai 264005

Received:2008-12-08 Revised:2009-12-14 Online:2010-12-25 Published:2010-12-25
Supported by:
国家自然科学基金(10771117)和山东省青年自然科学基金(Q2007A02, ZR2009AL016)资助

摘要/Abstract

摘要：

该文研究一类具有变号非线性项的m -点边值问题
$$ \left\{\begin{array}{l}(p(t)u'(t))'-q(t)u(t)+ h(t)f(t,u(t))=0,\quad 0 \disp cu(1)+dp(1)u'(1)=\sum_{i=1}^{m-2}\beta_{i}u(\xi_{i}), \end{array}\right.$$
其中$f\in C([0,1]\times (-\infty,+\infty),(-\infty,+\infty))$, $f$不要求非负和下方有界. 通过建立更一般的Leray-Schauder度理论和计算全连续域上的拓扑度, 得到了非平凡解的存在性结果.

关键词: 非平凡解, m -点边值问题, 奇异, Leray-Schauder度

Abstract:

In this paper, the authors consider a class of m-point boundary value problems with changing sign nonlinearity $$ \left\{\begin{array}{l}
(p(t)u'(t))'-q(t)u(t)+ h(t)f(t,u(t))=0,\quad 0 where f ∈ C([0,1]×(-∞,+∞),(-∞,+∞)) is a sign-changing function, not necessarily nonnegative and bounded from below. By establishing a more general Leray-Schauder degree theory, and computing the topological degree of a completely continuous field, some existence results of nontrival solutions are obtained.

Key words: Nontrivial solutions, m-point boundary value problems, Singular, Leray-Schauder degree

中图分类号:

34B15| 34B25

栾世霞, 万菲菲, 张新光. 具有变号非线性项的多点边值问题的非平凡解[J]. 数学物理学报, 2010, 30(6): 1503-1513.

LUAN Shi-Xia, WAN Fei-Fei, ZHANG Xin-Guang. Nontrivial Solutions of Multi-point Boundary Value Problems with Sign-changing Nonlinear Terms[J]. Acta mathematica scientia,Series A, 2010, 30(6): 1503-1513.

参考文献

[1] Agarwal R P, Regan D O. A note on existence of nonnegative solutions to singular semi-positone problems. Nonlinear Anal, 1999, 36(5): 615--622

[2] 程建纲. 二阶边值问题的正解. 数学学报, 2001, 44(3): 429--436

[3] Erbe L H, Mathsen R M. Positive solutions for singular nonlinear boundary value problems. Nonlinear Anal, 2001, 46(7): 979--986

[4] 刘嘉荃. 一类奇异二阶边值问题的正解. 曲阜师范大学学报, 2002, 28(4): 1--7

[5] 赵增勤. 非线性奇异微分方程边值问题的正解. 数学学报, 2000, 43(1): 179--188

[6] Erbe L H, Hu S C, Wang H Y. Multiple positive solutions of some boundary value problems. J Math Anal Appl, 1994, 184(3): 640--648

[7] Henderson J, Wang H Y. Positive solutions for nonlinear eigenvalue problems. J Math Anal Appl, 1997, 208: 252--259

[8] Lan K Q, Webb J R. Positive solutions of semilinear differential equations with singularities. J Differential Equations, 1998, 148(2): 407--421

[9] Ma R Y. Multiple positive solutions for nonlinear $m$-point boundary value problems. Appl Math Comput, 2004, 148: 249--262

[10] Ma R Y. Positive solutions for second-order m-point boundary value problems of Robin type. Acta Mathematiea Scientia, 2004, 24(3): 307--318

[11] Ma R Y, Thompson B. Positive solutions for nonlinear m-point eigenvalue problems. J Math Anal Appl, 2004, 297(1): 24--37

[12] Han G D, Wu Y. Nontrivial solutions for singular two-point boundary value problems with sign-changing nonlinear term. J Math Anal Appl, 2007, 325(2): 1327--1338

[13] Li F Y, Han G D. Existence of non-zero solutions to nonlinear Hammerstein integral equation. 山西大学学报, 2003, 26(4): 283--286

[14] Sun J X, Zhang G W. Nontrivial solutions of singular superlinear Sturn-Liouville problem. J Math Anal Appl, 2006, 313: 518--536

[15] Deimling K. Nonlinear Functional Analysis. Berlin: Springer-Verlag, 1985

[16] Guo D J, Lakshmikantham V. Nonlinear Problems in Abstract Cones. New York: Academic Press, 1988

[17] 郭大钧, 孙经先. 非线性积分方程. 济南:山东科技出版社, 1987

[18] Zhang X G, Liu L S. Positive solutions of fourth-order multi-point boundary value problems with bending term. Appl Math Comput, 2007, 194: 321--332

[19] Zhang X G. Eigenvalue of higher-order semipositone multi-point boundary value problems with derivatives. Appl Math Comput, 2008, 201: 361--370