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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 |
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 $.
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. |
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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.
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[5] |
J. Byeon, O. 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 Nezza, G. 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. Fall, F. 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. He, G. 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. Yu, F. 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. Zhang, J. 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. Byeon, O. 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 Nezza, G. 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. Fall, F. 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. He, G. 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. Yu, F. 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. Zhang, J. 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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