Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent

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January 2018, 17(1): 113-125. doi: 10.3934/cpaa.2018007

Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent

Peng Chen and Xiaochun Liu ^,

School of mathematics and statistics, Wuhan University, Wuhan, 430072, China

* Corresponding author

Received January 2017 Revised July 2017 Published September 2017

Fund Project: The authors are supported by NSFC grants 11371282 and 11571259

This paper is concerned with the existence and multiplicity of solutions to the following Kirchhoff type elliptic equations with critical nonlinearity:

$ \begin{cases} -(a+b \int_{Ω}|\nabla u|^{2}dx)Δ u=f(x, u)+μ|u|^{4}u &\; \; \mbox{in }Ω, \\ u=0 &\; \; \mbox{on }\partial Ω, \end{cases}$

where $Ω\subset\mathbb{R}^3$ is a bounded smooth domain, $μ$ is a positive parameter and $f:Ω×\mathbb{R}\to \mathbb{R}$ is a Carathéodory function satisfying some further conditions. Our approach is based on concentration-compactness principle and symmetry mountain pass theorem.

Keywords: Kirchhoff type equation, multiplicity, critical nonlinearity.

Mathematics Subject Classification: Primary:35B33, 35J60;Secondary:47J30.

Citation: Peng Chen, Xiaochun Liu. Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2018, 17 (1) : 113-125. doi: 10.3934/cpaa.2018007

References:

[1]	A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 41 (1973), 349-381.
[2]	C. O. Alves, F. J. S. A. Corra and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93. doi: 10.1016/j.camwa.2005.01.008.
[3]	C. O. Alves, F. J. S. A. Corra and G. M. Figueiredo, On a class of nonlocal elliptic problems with critical growth, Differ. Equ. Appl., 23 (2010), 409-417. doi: 10.7153/dea-02-25.
[4]	H. Brézis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.
[5]	G. M. Figueiredo and J. R. Santos Junior, Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth, Differential Integral Equations, 25 (2012), 853-868.
[6]	G. M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl., 401 (2013), 706-713. doi: 10.1016/j.jmaa.2012.12.053.
[7]	C. Heil, A Basis Theory Primer Expanded edition, Applied and Numerical Harmonic Analysis, Birkhäuser, New York, 2011. doi: 10.1007/978-0-8176-4687-5.
[8]	X. M. He and W. M. Zou, Infinitely many positive solutions for Kirchhoff type problems, Nonlinear Anal., 70 (2009), 1407-1414. doi: 10.1016/j.na.2008.02.021.
[9]	G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
[10]	J. L. Lions, On some questions in boundary value problems of mathematical physics, in Contemporary Development in Continuum Mechanics and Partial Differential Equations, in: North-Holland Math. Stud., vol. 30, North-Holland, Amsterdam, New York, 1978, pp. 284-346.
[11]	P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, Part 1, Rev. Mat. Iberoamericana, 1 (1985), 145-201. doi: 10.4171/RMI/6.
[12]	Z. Liang, F. Li and J. Shi, Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 155-167. doi: 10.1016/j.anihpc.2013.01.006.
[13]	A. Mao and Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 70 (2009), 1275-1287. doi: 10.1016/j.na.2008.02.011.
[14]	D. Naimen, Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 885-914. doi: 10.1007/s00030-014-0271-4.
[15]	K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff type problems via the Yang index, J. Differential Equations, 221 (2006), 246-255. doi: 10.1016/j.jde.2005.03.006.
[16]	E. A. B. Silva and M. S. Xavier, Multiplicity of solutions for quasilinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 341-358. doi: 10.1016/S0294-1449(02)00013-6.
[17]	J. J. Sun and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212-1222. doi: 10.1016/j.na.2010.09.061.
[18]	Q. L. Xie, X. P. Wu and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent, Commun. Pure Appl. Anal., 12 (2013), 2773-2786. doi: 10.3934/cpaa.2013.12.2773.
[19]	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]	A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 41 (1973), 349-381.
[2]	C. O. Alves, F. J. S. A. Corra and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93. doi: 10.1016/j.camwa.2005.01.008.
[3]	C. O. Alves, F. J. S. A. Corra and G. M. Figueiredo, On a class of nonlocal elliptic problems with critical growth, Differ. Equ. Appl., 23 (2010), 409-417. doi: 10.7153/dea-02-25.
[4]	H. Brézis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.
[5]	G. M. Figueiredo and J. R. Santos Junior, Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth, Differential Integral Equations, 25 (2012), 853-868.
[6]	G. M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl., 401 (2013), 706-713. doi: 10.1016/j.jmaa.2012.12.053.
[7]	C. Heil, A Basis Theory Primer Expanded edition, Applied and Numerical Harmonic Analysis, Birkhäuser, New York, 2011. doi: 10.1007/978-0-8176-4687-5.
[8]	X. M. He and W. M. Zou, Infinitely many positive solutions for Kirchhoff type problems, Nonlinear Anal., 70 (2009), 1407-1414. doi: 10.1016/j.na.2008.02.021.
[9]	G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
[10]	J. L. Lions, On some questions in boundary value problems of mathematical physics, in Contemporary Development in Continuum Mechanics and Partial Differential Equations, in: North-Holland Math. Stud., vol. 30, North-Holland, Amsterdam, New York, 1978, pp. 284-346.
[11]	P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, Part 1, Rev. Mat. Iberoamericana, 1 (1985), 145-201. doi: 10.4171/RMI/6.
[12]	Z. Liang, F. Li and J. Shi, Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 155-167. doi: 10.1016/j.anihpc.2013.01.006.
[13]	A. Mao and Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 70 (2009), 1275-1287. doi: 10.1016/j.na.2008.02.011.
[14]	D. Naimen, Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 885-914. doi: 10.1007/s00030-014-0271-4.
[15]	K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff type problems via the Yang index, J. Differential Equations, 221 (2006), 246-255. doi: 10.1016/j.jde.2005.03.006.
[16]	E. A. B. Silva and M. S. Xavier, Multiplicity of solutions for quasilinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 341-358. doi: 10.1016/S0294-1449(02)00013-6.
[17]	J. J. Sun and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212-1222. doi: 10.1016/j.na.2010.09.061.
[18]	Q. L. Xie, X. P. Wu and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent, Commun. Pure Appl. Anal., 12 (2013), 2773-2786. doi: 10.3934/cpaa.2013.12.2773.
[19]	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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