September  2021, 20(9): 3065-3092. doi: 10.3934/cpaa.2021096

Multiplicity and concentration of positive solutions to the fractional Kirchhoff type problems involving sign-changing weight functions

1. 

School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, 410083, China

2. 

School of Mathematics and Computational Science, Huaihua University, Huaihua, 418008, China

* Corresponding author

Received  November 2020 Revised  May 2021 Published  September 2021 Early access  June 2021

Fund Project: This work was supported by the National Natural Science Foundation of China (No. 12071486), the Research Foundation of Education Bureau of Hunan Province, China (No. 20B457, 19B450, 20A387)

The aim of this paper is to study the multiplicity and concentration of positive solutions to the fractional Kirchhoff type problems involving sign-changing weight functions and concave-convex nonlinearities with subcritical or critical growth. Applying Nehari manifold, fibering maps and Ljusternik-Schnirelmann theory, we investigate a relationship between the number of positive solutions and the topology of the global maximum set of $ K $.

Citation: Jie Yang, Haibo Chen. Multiplicity and concentration of positive solutions to the fractional Kirchhoff type problems involving sign-changing weight functions. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3065-3092. doi: 10.3934/cpaa.2021096
References:
[1]

C. O. Alves and V. Ambrosio, Existence, multiplicity and concentration for a class of fractional $p \& q$ Laplacian problems in $\mathbb{R}^{N}$, Commun. Pure Appl. Anal., 18 (2019), 2009-2045.  doi: 10.3934/cpaa.2019091.

[2]

K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differ. Equ., 193 (2003), 481-499.  doi: 10.1016/S0022-0396(03)00121-9.

[3]

V. Benci and G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differ. Equ., 2 (1994), 29-48.  doi: 10.1007/BF01234314.

[4]

H. Brezis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, P. Am. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.

[5]

J. ByeonO. Kwon and J. Seok, Nonlinear scalar field equations involving the fractional Laplacian, Nonlinearity, 30 (2017), 1659-1681.  doi: 10.1088/1361-6544/aa60b4.

[6]

G. F. Che and H. B. Chen., Existence and asymptotic behavior of positive ground state solutions for coupled nonlinear fractional Kirchhoff-type systems, Comput. Math. Appl., 77 (2019), 173-188.  doi: 10.1016/j.camwa.2018.09.020.

[7]

A. Cotsiolis and N. K. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.  doi: 10.1016/j.jmaa.2004.03.034.

[8]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates, Ann. I. H. Poincare-An., 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001.

[9]

C. Y. Chen and T. F. Wu, Multiple positive solutions for indefinite semilinear elliptic problems involving a critical Sobolev exponent, P. Roy. Soc. Edinb. A, 144 (2014), 691-709.  doi: 10.1017/S0308210512000133.

[10]

R. Dr$\acute{a}$bek and S. I. Pohozaev, Positive solutions for the $p$-Laplacian: Application of the fibering method, P. Roy. Soc. Edinb. A, 127 (1997), 703-726.  doi: 10.1017/S0308210500023787.

[11]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[12]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.

[13]

A. Fiscella and P. K. Mishra, The Nehari manifold for fractional Kirchhoff problems involving singular and critical terms, Nonlinear Anal., 186 (2019), 6-32.  doi: 10.1016/j.na.2018.09.006.

[14]

M. M. FallF. Mahmoudi and E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961.  doi: 10.1088/0951-7715/28/6/1937.

[15]

X. M. He and W. M. Zou, Ground states for nonlinear Kirchhoff equations with critical growth, Ann. Mat. Pur. Appl., 193 (2014), 473-500.  doi: 10.1007/s10231-012-0286-6.

[16]

Y. He, Concentrating bounded states for a class of singularlyperturbed Kirchhoff type equations with ageneral nonlinearity, J. Differ. Equ., 261 (2016), 6178-6220.  doi: 10.1016/j.jde.2016.08.034.

[17]

Y. He and G. B. Li, Standing waves for a class of Kirchhoff type problems in $\mathbb{R}^{3}$ involving critical Sobolev exponents, Calc. Var. Partial Differ. Equ., 54 (2015), 3067-3106.  doi: 10.1007/s00526-015-0894-2.

[18]

Y. HeG. B. Li and S. J. Peng, Concentrating bound states for Kirchhoff type Problems in $\mathbb{R}^{3}$ involving critical Sobolev exponents, Adv. Nonlinear Stud., 14 (2014), 483-510.  doi: 10.1515/ans-2014-0214.

[19]

X. M. He and W. M. Zou, Multiplicity of concentrating solutions for a class of fractional Kirchhoff equation, Manuscripta Math., 158 (2018), 159-203.  doi: 10.1007/s00229-018-1017-0.

[20]

S. L. Liu, H. B. Chen and J. Yang, Existence and nonexistence of solutions for a class of Kirchhoff type equation involving fractional $p$-Laplacian, Racsam. Rev. R. Acad. A, 114 (2020), Art. 161. doi: 10.1007/s13398-020-00893-5.

[21]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. I. H. Poincare-An., 1 (1984), 109-145. 

[22]

G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differ. Equ., 50 (2014), 799-829.  doi: 10.1007/s00526-013-0656-y.

[23]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, T. Am. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.

[24]

S. Secchi, Ground state solutions for nonlinear Schrödinger equations in $\mathbb{R}^{3}$, J Math Phys., 54 (2013), Art. 031501. doi: 10.1063/1.4793990.

[25]

X. Shang and J. Zhang, Existence and multiplicity solutions of fractional Schrödinger equation with competing potential functions, Complex Var. Elliptic, 61 (2016), 1435-1463.  doi: 10.1080/17476933.2016.1182516.

[26]

X. Shang and J. Zhang, Concentrating solutions of nonlinear fractional Schrödinger equation with potentials, J. Differ. Equ., 258 (2015), 1106-1128.  doi: 10.1016/j.jde.2014.10.012.

[27]

Y. Su and H. B. Chen, Fractional Kirchhoff-type equation with Hardy-Littlewood-Sobolev critical exponent, Comput. Math. Appl., 78 (2019), 2063-2082.  doi: 10.1016/j.camwa.2019.03.052.

[28]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.

[29]

T. F. Wu, Multiple positive solutions for a class of concave-convex elliptic problems in $\mathbb{R}^{N}$ involving sign-changing weight, J. Funct. Anal., 258 (2010), 99-131.  doi: 10.1016/j.jfa.2009.08.005.

[30]

Y. Wei and X. Su, Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian, Calc. Var. Partial Differ. Equ., 52 (2015), 95-124.  doi: 10.1007/s00526-013-0706-5.

[31]

W. H. Xie and H. B. Chen, Multiple positive solutions for the critical Kirchhoff type problems involving sign-changing weight functions, J. Math. Anal. Appl., 479 (2019), 135-161.  doi: 10.1016/j.jmaa.2019.06.020.

[32]

W. H. Xie and H. B. Chen, On the Kirchhoff problems involving critical Sobolev exponent, Appl. Math. Lett., 105 (2020), Art. 106346. doi: 10.1016/j.aml.2020.106346.

[33]

Y. YuF. Zhao and L. Zhao, The existence and multiplicity of solutions of a fractional Schrödinger-Poisson system with critical growth, Sci. China Math., 61 (2018), 1039-1062.  doi: 10.1007/s11425-016-9074-6.

[34]

J. Yang, H. B. Chen and Z. S. Feng, Multiple positive solutions to the fractional Kirchhoff problem with critical indefinite nonlinearities, Electron. J. Differ. Equ., 2020 (2020), Art. 101.

[35]

J. ZhangJ. C. Wang and Y. J. Ji, The critical fractional Schrödinger equation with a small superlinear term, Nonlinear Anal-Real., 45 (2019), 200-225.  doi: 10.1016/j.nonrwa.2018.07.003.

[36]

J. Zhang, J. T. Sun and T. F. Wu, The number of positive solutions affected by the weight function to Kirchhoff type equations in high dimensions, Nonlinear Anal-Theor., 196 (2020), Art. 111780. doi: 10.1016/j.na.2020.111780.

show all references

References:
[1]

C. O. Alves and V. Ambrosio, Existence, multiplicity and concentration for a class of fractional $p \& q$ Laplacian problems in $\mathbb{R}^{N}$, Commun. Pure Appl. Anal., 18 (2019), 2009-2045.  doi: 10.3934/cpaa.2019091.

[2]

K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differ. Equ., 193 (2003), 481-499.  doi: 10.1016/S0022-0396(03)00121-9.

[3]

V. Benci and G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differ. Equ., 2 (1994), 29-48.  doi: 10.1007/BF01234314.

[4]

H. Brezis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, P. Am. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.

[5]

J. ByeonO. Kwon and J. Seok, Nonlinear scalar field equations involving the fractional Laplacian, Nonlinearity, 30 (2017), 1659-1681.  doi: 10.1088/1361-6544/aa60b4.

[6]

G. F. Che and H. B. Chen., Existence and asymptotic behavior of positive ground state solutions for coupled nonlinear fractional Kirchhoff-type systems, Comput. Math. Appl., 77 (2019), 173-188.  doi: 10.1016/j.camwa.2018.09.020.

[7]

A. Cotsiolis and N. K. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.  doi: 10.1016/j.jmaa.2004.03.034.

[8]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates, Ann. I. H. Poincare-An., 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001.

[9]

C. Y. Chen and T. F. Wu, Multiple positive solutions for indefinite semilinear elliptic problems involving a critical Sobolev exponent, P. Roy. Soc. Edinb. A, 144 (2014), 691-709.  doi: 10.1017/S0308210512000133.

[10]

R. Dr$\acute{a}$bek and S. I. Pohozaev, Positive solutions for the $p$-Laplacian: Application of the fibering method, P. Roy. Soc. Edinb. A, 127 (1997), 703-726.  doi: 10.1017/S0308210500023787.

[11]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[12]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.

[13]

A. Fiscella and P. K. Mishra, The Nehari manifold for fractional Kirchhoff problems involving singular and critical terms, Nonlinear Anal., 186 (2019), 6-32.  doi: 10.1016/j.na.2018.09.006.

[14]

M. M. FallF. Mahmoudi and E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961.  doi: 10.1088/0951-7715/28/6/1937.

[15]

X. M. He and W. M. Zou, Ground states for nonlinear Kirchhoff equations with critical growth, Ann. Mat. Pur. Appl., 193 (2014), 473-500.  doi: 10.1007/s10231-012-0286-6.

[16]

Y. He, Concentrating bounded states for a class of singularlyperturbed Kirchhoff type equations with ageneral nonlinearity, J. Differ. Equ., 261 (2016), 6178-6220.  doi: 10.1016/j.jde.2016.08.034.

[17]

Y. He and G. B. Li, Standing waves for a class of Kirchhoff type problems in $\mathbb{R}^{3}$ involving critical Sobolev exponents, Calc. Var. Partial Differ. Equ., 54 (2015), 3067-3106.  doi: 10.1007/s00526-015-0894-2.

[18]

Y. HeG. B. Li and S. J. Peng, Concentrating bound states for Kirchhoff type Problems in $\mathbb{R}^{3}$ involving critical Sobolev exponents, Adv. Nonlinear Stud., 14 (2014), 483-510.  doi: 10.1515/ans-2014-0214.

[19]

X. M. He and W. M. Zou, Multiplicity of concentrating solutions for a class of fractional Kirchhoff equation, Manuscripta Math., 158 (2018), 159-203.  doi: 10.1007/s00229-018-1017-0.

[20]

S. L. Liu, H. B. Chen and J. Yang, Existence and nonexistence of solutions for a class of Kirchhoff type equation involving fractional $p$-Laplacian, Racsam. Rev. R. Acad. A, 114 (2020), Art. 161. doi: 10.1007/s13398-020-00893-5.

[21]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. I. H. Poincare-An., 1 (1984), 109-145. 

[22]

G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differ. Equ., 50 (2014), 799-829.  doi: 10.1007/s00526-013-0656-y.

[23]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, T. Am. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.

[24]

S. Secchi, Ground state solutions for nonlinear Schrödinger equations in $\mathbb{R}^{3}$, J Math Phys., 54 (2013), Art. 031501. doi: 10.1063/1.4793990.

[25]

X. Shang and J. Zhang, Existence and multiplicity solutions of fractional Schrödinger equation with competing potential functions, Complex Var. Elliptic, 61 (2016), 1435-1463.  doi: 10.1080/17476933.2016.1182516.

[26]

X. Shang and J. Zhang, Concentrating solutions of nonlinear fractional Schrödinger equation with potentials, J. Differ. Equ., 258 (2015), 1106-1128.  doi: 10.1016/j.jde.2014.10.012.

[27]

Y. Su and H. B. Chen, Fractional Kirchhoff-type equation with Hardy-Littlewood-Sobolev critical exponent, Comput. Math. Appl., 78 (2019), 2063-2082.  doi: 10.1016/j.camwa.2019.03.052.

[28]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.

[29]

T. F. Wu, Multiple positive solutions for a class of concave-convex elliptic problems in $\mathbb{R}^{N}$ involving sign-changing weight, J. Funct. Anal., 258 (2010), 99-131.  doi: 10.1016/j.jfa.2009.08.005.

[30]

Y. Wei and X. Su, Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian, Calc. Var. Partial Differ. Equ., 52 (2015), 95-124.  doi: 10.1007/s00526-013-0706-5.

[31]

W. H. Xie and H. B. Chen, Multiple positive solutions for the critical Kirchhoff type problems involving sign-changing weight functions, J. Math. Anal. Appl., 479 (2019), 135-161.  doi: 10.1016/j.jmaa.2019.06.020.

[32]

W. H. Xie and H. B. Chen, On the Kirchhoff problems involving critical Sobolev exponent, Appl. Math. Lett., 105 (2020), Art. 106346. doi: 10.1016/j.aml.2020.106346.

[33]

Y. YuF. Zhao and L. Zhao, The existence and multiplicity of solutions of a fractional Schrödinger-Poisson system with critical growth, Sci. China Math., 61 (2018), 1039-1062.  doi: 10.1007/s11425-016-9074-6.

[34]

J. Yang, H. B. Chen and Z. S. Feng, Multiple positive solutions to the fractional Kirchhoff problem with critical indefinite nonlinearities, Electron. J. Differ. Equ., 2020 (2020), Art. 101.

[35]

J. ZhangJ. C. Wang and Y. J. Ji, The critical fractional Schrödinger equation with a small superlinear term, Nonlinear Anal-Real., 45 (2019), 200-225.  doi: 10.1016/j.nonrwa.2018.07.003.

[36]

J. Zhang, J. T. Sun and T. F. Wu, The number of positive solutions affected by the weight function to Kirchhoff type equations in high dimensions, Nonlinear Anal-Theor., 196 (2020), Art. 111780. doi: 10.1016/j.na.2020.111780.

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