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March  2013, 12(2): 721-733. doi: 10.3934/cpaa.2013.12.721

Multiple positive solutions for Kirchhoff type problems with singularity

Xing Liu 1, and  Yijing Sun 2,

1. 

Department of Mathematics, University of Chinese Academy of Sciences, Beijing 100049, P.R. China
2. 

Department of Mathematics, Graduate University of Chinese Academy of Sciences, Beijing 100049, China

Received  July 2011 Revised  May 2012 Published  September 2012

A class of Kirchhoff type problems containing both singular and superlinear terms is considered in a bounded domain in $R^3$: multiplicity results are obtained by variational methods.
Keywords: Kirchhoff type equation, negative exponent., Nehari manifold.
Mathematics Subject Classification: Primary: 35J75, 35J20; Secondary: 35J6.
Citation: Xing Liu, Yijing Sun. Multiple positive solutions for Kirchhoff type problems with singularity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 721-733. doi: 10.3934/cpaa.2013.12.721
References:
[1]

G. Anello, A uniqueness result for a nonlocal equation of Kirchhoff type and some related open problems, J. Math. Anal. Appl., 373 (2011), 248-251. doi: 10.1016/j.jmaa.2010.07.019.  Google Scholar

[2]

C. Chen, Y. Kuo and T. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations, 250 (2011), 1876-1908. doi: 10.1016/j.jde.2010.11.017.  Google Scholar

[3]

J. Graham-Eagle, A variational approach to upper and lower solutions, IMA J. Appl. Math., 44 (1990), 181-184. doi: 10.1093/imamat/44.2.181.  Google Scholar

[4]

X. He and W. Zou, Infinitely many solutions for Kirchhoff-type problems, Nonlinear Anal., 70 (2009), 1407-1414. doi: 10.1016/j.na.2008.02.021.  Google Scholar

[5]

D. S. Kang, On the quasilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy terms, Nonlinear Anal., 68 (2008), 1973-1985. doi: 10.1016/j.na.2007.01.024.  Google Scholar

[6]

B. Ricceri, On an elliptic Kirchhoff-type problem depending on two parameters, J. Global Optim., 46 (2010), 543-549. doi: 10.1007/s10898-009-9438-7.  Google Scholar

[7]

Y. J. Sun and S. J. Li, Some remarks on a superlinear-singular problem: Estimates for $\lambda^*$, Nonlinear Anal., 69 (2008), 2636-2650. doi: 10.1016/j.na.2007.08.037.  Google Scholar

[8]

Y. J. Sun, S. P. Wu and Y. M. Long, Combined effects of singular and superlinear nonlinearities in some singular boundary value problems, J. Differential Equations, 176 (2001), 511-531. doi: 10.1006/jdeq.2000.3973.  Google Scholar

[9]

Y. J. Sun and S. P. Wu, An exact estimate result for a class of singular equations with critical exponents, J. Funct. Anal., 260 (2011), 1257-1284. doi: 10.1016/j.jfa.2010.11.018.  Google Scholar

[10]

G. Talenti, Best constant in Sobolev inequality, Ann. Math. Pure Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.  Google Scholar

[11]

Z. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317 (2006), 456-463. doi: 10.1016/j.jmaa.2005.06.102.  Google Scholar

show all references

References:
[1]

G. Anello, A uniqueness result for a nonlocal equation of Kirchhoff type and some related open problems, J. Math. Anal. Appl., 373 (2011), 248-251. doi: 10.1016/j.jmaa.2010.07.019.  Google Scholar

[2]

C. Chen, Y. Kuo and T. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations, 250 (2011), 1876-1908. doi: 10.1016/j.jde.2010.11.017.  Google Scholar

[3]

J. Graham-Eagle, A variational approach to upper and lower solutions, IMA J. Appl. Math., 44 (1990), 181-184. doi: 10.1093/imamat/44.2.181.  Google Scholar

[4]

X. He and W. Zou, Infinitely many solutions for Kirchhoff-type problems, Nonlinear Anal., 70 (2009), 1407-1414. doi: 10.1016/j.na.2008.02.021.  Google Scholar

[5]

D. S. Kang, On the quasilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy terms, Nonlinear Anal., 68 (2008), 1973-1985. doi: 10.1016/j.na.2007.01.024.  Google Scholar

[6]

B. Ricceri, On an elliptic Kirchhoff-type problem depending on two parameters, J. Global Optim., 46 (2010), 543-549. doi: 10.1007/s10898-009-9438-7.  Google Scholar

[7]

Y. J. Sun and S. J. Li, Some remarks on a superlinear-singular problem: Estimates for $\lambda^*$, Nonlinear Anal., 69 (2008), 2636-2650. doi: 10.1016/j.na.2007.08.037.  Google Scholar

[8]

Y. J. Sun, S. P. Wu and Y. M. Long, Combined effects of singular and superlinear nonlinearities in some singular boundary value problems, J. Differential Equations, 176 (2001), 511-531. doi: 10.1006/jdeq.2000.3973.  Google Scholar

[9]

Y. J. Sun and S. P. Wu, An exact estimate result for a class of singular equations with critical exponents, J. Funct. Anal., 260 (2011), 1257-1284. doi: 10.1016/j.jfa.2010.11.018.  Google Scholar

[10]

G. Talenti, Best constant in Sobolev inequality, Ann. Math. Pure Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.  Google Scholar

[11]

Z. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317 (2006), 456-463. doi: 10.1016/j.jmaa.2005.06.102.  Google Scholar

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