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.

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.

[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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