MULTIPLICITY AND CONCENTRATION OF SOLUTIONS TO A SINGULAR CHOQUARD EQUATION WITH CRITICAL SOBOLEV EXPONENT

Shengbin Yu; Jianqing Chen

doi:10.11948/20230439

2025 Volume 15 Issue 1

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Shengbin Yu, Jianqing Chen. MULTIPLICITY AND CONCENTRATION OF SOLUTIONS TO A SINGULAR CHOQUARD EQUATION WITH CRITICAL SOBOLEV EXPONENT[J]. Journal of Applied Analysis & Computation, 2025, 15(1): 110-136. doi: 10.11948/20230439

Citation:

Shengbin Yu, Jianqing Chen. MULTIPLICITY AND CONCENTRATION OF SOLUTIONS TO A SINGULAR CHOQUARD EQUATION WITH CRITICAL SOBOLEV EXPONENT[J]. Journal of Applied Analysis & Computation, 2025, 15(1): 110-136. doi: 10.11948/20230439

MULTIPLICITY AND CONCENTRATION OF SOLUTIONS TO A SINGULAR CHOQUARD EQUATION WITH CRITICAL SOBOLEV EXPONENT

Shengbin Yu^1, ,,
Jianqing Chen^2,

1.
College of Information Engineering, Fujian Business University, Fuzhou 350102, China
2.
School of Mathematics and Statistics & FJKLMAA, Fujian Normal University, Qishan Campus, Fuzhou 350117, China

Author Bio: Email: jqchen@fjnu.edu.cn(J. Chen)
Corresponding author: Email: yushengbin.8@163.com(S. Yu)
Fund Project: The authors were supported by National Natural Science Foundation of China (No. 11871152, 11671085) and Natural Science Foundation of Fujian Province (No. 2023J01163)

Abstract

Abstract

In this paper, we consider a nonautonomous singular Choquard equation with critical exponent
$ \left \{\begin{array}{lcl} -\Delta u+V(x)u+\lambda(I_\alpha\ast |u|^{p})|u|^{p-2}u=f(x)u^{-\gamma}+|u|^{4}u, &&\quad x\in\mathbb{R}^3,\\ u>0,&&\quad x\in\mathbb{R}^3, \end{array}\right. $
where $ I_\alpha $ is the Riesz potential of order $ \alpha\in(0,3) $ and $ 1+\frac{\alpha}{3}\le p<3 $, $ 0<\gamma<1 $. Under certain assumptions on $ V $ and $ f $, we show the existence and multiplicity of positive solutions for $ \lambda>0 $ by using variational method and Nehari type constraint. We also study concentration of solutions as $ \lambda\rightarrow0^+ $.
- Singular Choquard equation /
- variational method /
- concentration /
- critical Sobolev exponent.
MSC: 35J20, 35J75, 35B09, 35B40

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References

[1]	Y. Ao, Existence of solutions for a class of nonlinear Choquard equations with critical growth, Appl. Anal., 2021, 100(3), 465-481. doi: 10.1080/00036811.2019.1608961 CrossRef Google Scholar
[2]	T. Bartsch and Z. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^N$, Comm. Partial Differential Equations, 1995, 20(9-10), 1725-1741. doi: 10.1080/03605309508821149 CrossRef Google Scholar
[3]	H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 1983, 36(4), 437-477. doi: 10.1002/cpa.3160360405 CrossRef Google Scholar
[4]	M. Coclite and G. Palmieri, On a singular nonlinear Dirichlet problem, Comm. Partial Differential Equations, 1989, 14(10), 1315-1327. doi: 10.1080/03605308908820656 CrossRef Google Scholar
[5]	A. Fiscella and P. Mishra, The Nehari manifold for fractional Kirchhoff problems involving singular and critical terms, Nonlinear Anal., 2019, 186, 6-32. doi: 10.1016/j.na.2018.09.006 CrossRef Google Scholar
[6]	M. Ghimenti, V. Moroz and J. Schaftingen, Least action nodal solutions for the quadratic Choquard equation, Proc. Amer. Math. Soc., 2017, 145, 737-747. Google Scholar
[7]	J. Giacomoni and K. Saoudi, Multiplicity of positive solutions for a singular and critical problem, Nonlinear Anal., 2009, 71(9), 4060-4077. doi: 10.1016/j.na.2009.02.087 CrossRef Google Scholar
[8]	N. Hirano, C. Saccon and N. Shioji, Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities, Adv. Differential Equations, 2004, 9(1-2), 197-220. Google Scholar
[9]	N. Hirano, C. Saccon and N. Shioji, Brezis-Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem, J. Differential Equations, 2008, 245(8), 1997-2037. doi: 10.1016/j.jde.2008.06.020 CrossRef Google Scholar
[10]	L. Huang, E. Rocha and J. Chen, Positive and sign-changing solutions of a Schrödinger-Poisson system involving a critical nonlinearity, J. Math. Anal. Appl., 2013, 408(1), 55-69. doi: 10.1016/j.jmaa.2013.05.071 CrossRef Google Scholar
[11]	C. Lei and J. Liao, Multiple positive solutions for Schrödinger-Poisson system involving singularity and critical exponent, Math. Meth. Appl. Sci., 2019, 42(7), 2417-2430. doi: 10.1002/mma.5519 CrossRef Google Scholar
[12]	C. Lei, J. Liao and C. Tang, Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents, J. Math. Anal. Appl., 2015, 421(1), 521-538. doi: 10.1016/j.jmaa.2014.07.031 CrossRef Google Scholar
[13]	C. Lei, H. Suo and C. Chu, Multiple positive solutions for a Schrödinger-Newton system with singularity and critical growth, Electron. J. Differential Equations, 2018, 86, 1-15. Google Scholar
[14]	F. Li, C. Gao and X. Zhu, Existence and concentration of sign-changing solutions to Kirchhoff-type system with Hartree-type nonlinearity, J. Math. Anal. Appl., 2017, 448(1), 60-80. doi: 10.1016/j.jmaa.2016.10.069 CrossRef Google Scholar
[15]	G. Li, Y. Li, C. Tang and L. Yin, Existence and concentrate behavior of ground state solutions for critical Choquard equations, Appl. Math. Lett., 2019, 96, 101-107. doi: 10.1016/j.aml.2019.04.020 CrossRef Google Scholar
[16]	G. Li and C. Tang, Existence of a ground state solution for Choquard equation with the upper critical exponent, Comput. Math. Appl., 2018, 76(11-12), 2635-2647. doi: 10.1016/j.camwa.2018.08.052 CrossRef Google Scholar
[17]	X. Li and S. Ma, Choquard equations with critical nonlinearities, Commun. Contemp. Math., 2020, 22(4), 1950023. doi: 10.1142/S0219199719500238 CrossRef Google Scholar
[18]	X. Li, S. Ma and G. Zhang, Existence and qualitative properties of solutions for Choquard equations with a local term, Nonlinear Anal. Real World Appl., 2019, 45, 1-25. doi: 10.1016/j.nonrwa.2018.06.007 CrossRef Google Scholar
[19]	E. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 1977, 57(2), 93-105. doi: 10.1002/sapm197757293 CrossRef Google Scholar
[20]	J. Liu, A. Hou and J. Liao, Multiplicity of positive solutions for a class of singular elliptic equations with critical Sobolev exponent and Kirchhoff-type nonlocal term, Electron. J. Qual. Theory Differ. Equ., 2018, 100, 1-20. Google Scholar
[21]	D. Lü, Existence and concentration of solutions for a nonlinear Choquard equation, Mediterr. J. Math., 2015, 12(3), 839-850. doi: 10.1007/s00009-014-0428-8 CrossRef Google Scholar
[22]	D. Lü, A note on Kirchhoff-type equations with Hartree-type nonlinearities, Nonlinear Anal., 2014, 99, 35-48. doi: 10.1016/j.na.2013.12.022 CrossRef Google Scholar
[23]	C. Mercuri, V. Moroz and J. Schaftingen, Groundstates and radial solutions to nonlinear Schrödinger-Poisson-Slater equations at the critical frequency, Calc. Var. Partial Differ. Equ., 2016, 55, 146. doi: 10.1007/s00526-016-1079-3 CrossRef Google Scholar
[24]	V. Moroz and J. Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 2017, 19, 773-813. doi: 10.1007/s11784-016-0373-1 CrossRef Google Scholar
[25]	T. Mukherjee and K. Sreenadh, Positive solutions for nonlinear Choquard equation with singular nonlinearity, Complex Var. Elliptic Equ., 2017, 62(8), 1044-1071. doi: 10.1080/17476933.2016.1260559 CrossRef Google Scholar
[26]	S. Pekar, Untersuchungen über Die Elektronentheorie Der Kristalle, Akademie Verlag, Berlin, 1954. Google Scholar
[27]	D. Ruiz amd J. Schaftingen, Odd symmetry of least energy nodal solutions for the Choquard equation, J. Differential Equations, 2018, 264(2), 1231-1262. doi: 10.1016/j.jde.2017.09.034 CrossRef Google Scholar
[28]	J. Schaftingen and J. Xia, Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent, J. Math. Anal. Appl., 2018, 464(2), 1184-1202. doi: 10.1016/j.jmaa.2018.04.047 CrossRef Google Scholar
[29]	J. Seok, Limit profiles and uniqueness of ground states to the nonlinear Choquard equations, Adv. Nonlinear Anal., 2019, 8(1), 1083-1098. Google Scholar
[30]	J. Seok, Nonlinear Choquard equations involving a critical local term, Appl. Math. Lett., 2017, 63, 77-87. doi: 10.1016/j.aml.2016.07.027 CrossRef Google Scholar
[31]	J. Seok, Nonlinear Choquard equations: Doubly critical case, Appl. Math. Lett., 2018, 76, 148-156. doi: 10.1016/j.aml.2017.08.016 CrossRef Google Scholar
[32]	Y. Su and H. Chen, Existence of nontrivial solutions for a perturbation of Choquard equation with Hardy-Littlewood-Sobolev upper critical exponent, Electron. J. Differential Equations, 2018, 123, 1-25. Google Scholar
[33]	Y. Sun and S. Li, Structure of ground state solutions of singular semilinear elliptic equations, Nonlinear Anal., 2003, 55(4), 399-417. doi: 10.1016/S0362-546X(03)00244-X CrossRef Google Scholar
[34]	Y. Sun and S. Wu, An exact estimate result for a class of singular equations with critical exponents, J. Funct. Anal., 2011, 260(5), 1257-1284. doi: 10.1016/j.jfa.2010.11.018 CrossRef Google Scholar
[35]	X. Wang, L. Zhao and P. Zhao, Combined effects of singular and critical nonlinearities in elliptic problems, Nonlinear Anal., 2013, 87, 1-10. doi: 10.1016/j.na.2013.03.019 CrossRef Google Scholar
[36]	T. Wu, On a class of nonlocal nonlinear Schrödinger equations with potential well, Adv. Nonlinear Anal., 2020, 9(1), 665-689. Google Scholar
[37]	H. Yang, Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem, J. Differential Equations, 2003, 189(2), 487-512. doi: 10.1016/S0022-0396(02)00098-0 CrossRef Google Scholar
[38]	S. Yu and J. Chen, Uniqueness and asymptotical behavior of solutions to a Choquard equation with singularity, Appl. Math. Lett., 2020, 102, 106099. doi: 10.1016/j.aml.2019.106099 CrossRef Google Scholar
[39]	S. Yu and J. Chen, Multiple and asymptotical behavior of solutions to a Choquard equation with singularity, J. Math. Anal. Appl., 2022, 511, 126047. Google Scholar
[40]	S. Yu and J. Chen, Fractional Schrödinger-Poisson system with singularity: Existence, uniqueness and asymptotic behaviour, Glasgow Math. J., 2021, 63(1), 179-192. Google Scholar
[41]	S. Yu and J. Chen, Multiple positive solutions for critical elliptic problem with singularity, Monatsh. Math., 2021, 194, 395-423. Google Scholar
[42]	X. Zhong and C. Tang, Ground state sign-changing solutions for a class of subcritical Choquard equations with a critical pure power nonlinearity in $\mathbb{R}^N$, Comput. Math. Appl., 2018, 76(1), 23-34. Google Scholar