# American Institute of Mathematical Sciences

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

## Multiple positive solutions for Kirchhoff type problems with singularity

 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.
Citation: Xing Liu, Yijing Sun. Multiple positive solutions for Kirchhoff type problems with singularity. Communications on Pure and 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. [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. [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. [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. [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. [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. [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. [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. [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. [10] G. Talenti, Best constant in Sobolev inequality, Ann. Math. Pure Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013. [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.

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