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 & Applied Analysis, 2021, 20 (9) : 3065-3092. doi: 10.3934/cpaa.2021096
References:
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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.  Google Scholar

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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.  Google Scholar

[26]

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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