DOUBLE PHASE PROBLEM WITH SINGULARITY AND HOMOGENOUS CHOQUARD TYPE TERM

Omar Benslimane; Ahmed Aberqi; Mhamed Elmassoudi

doi:10.11948/20230303

2024 Volume 14 Issue 4

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2024 > 14(4): 2109-2124

Next Article Previous Article

Omar Benslimane, Ahmed Aberqi, Mhamed Elmassoudi. DOUBLE PHASE PROBLEM WITH SINGULARITY AND HOMOGENOUS CHOQUARD TYPE TERM[J]. Journal of Applied Analysis & Computation, 2024, 14(4): 2109-2124. doi: 10.11948/20230303

Citation:

Omar Benslimane, Ahmed Aberqi, Mhamed Elmassoudi. DOUBLE PHASE PROBLEM WITH SINGULARITY AND HOMOGENOUS CHOQUARD TYPE TERM[J]. Journal of Applied Analysis & Computation, 2024, 14(4): 2109-2124. doi: 10.11948/20230303

DOUBLE PHASE PROBLEM WITH SINGULARITY AND HOMOGENOUS CHOQUARD TYPE TERM

1.
EST of Salé, Mohammed V University, Rabat, Morocco
2.
Laboratory LAMA, ENSA, Sidi Mohamed Ben Abdellah University, Morocco
3.
Laboratory LAMA, FSDM, Sidi Mohamed Ben Abdellah University, Morocco

Author Bio: Email: aberqi_ahmed@yahoo.fr(A. Aberqi); Email: elmassoudi09@gmail.com(M. Elmassoudi)
Corresponding author: Email: omar.benslimane@usmba.ac.ma(O. Benslimane)

Abstract

Abstract

In this study, we prove in the context of Musielak Sobolev space that, under various assumptions on the data, two positive non-trivial solutions exist to the double phase problem with a singularity and a homogeneous Choquard type on the right-hand side. Our method relies on the Nehari manifold, the Hardy Littlewood - Sobolev inequality, and some variational approaches. The findings presented here generalize some known results.
- Double phase operator /
- singular problem /
- Choquard term /
- existence of solutions /
- variational method /
- Musielak-Orlicz spaces
MSC: 35A15, 35J20, 35J60, 35B38

FullText(HTML)

References

[1]	A. Aberqi, O. Benslimane, M. Elmassoudi and M. A. Ragusa, Nonnegative solution of a class of double phase problems with logarithmic nonlinearity, Boun. Val. Prob., 2022, 2022, 1–13. doi: 10.1186/s13661-021-01582-x CrossRef Google Scholar
[2]	A. Aberqi, O. Benslimane and M. Knifda, On a class of double phase problem involving potentials terms, J. Ellip. Parab. Equ., 2022, 8, 791–811. doi: 10.1007/s41808-022-00176-7 CrossRef Google Scholar
[3]	G. C. Anthal, J. Giacomoni and K. A. Sreenadh, Choquard-type equation with a singular absorption nonlinearity in two dimensions, Math. Meth. Appl. Sci., 2022, 46, 1–24. Google Scholar
[4]	R. Arora, A. Fiscella, T. Mukherjee and P. Winkert, Existence of ground state solutions for a Choquard double phase problem, Nonlinear Anal. : Real World Appl., 2023, 73, 103914. doi: 10.1016/j.nonrwa.2023.103914 CrossRef Google Scholar
[5]	O. Benslimane, A. Aberqi and J. Bennouna, Existence results for double phase obstacle problems with variable exponents, J. Ellip. Parab. Equ., 2021, 7, 850–890. Google Scholar
[6]	S. Cingolani, M. Gallo and K. Tanaka, Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities, Cal. Var. Par. Diff. Equ., 2022, 61, 1–34. Google Scholar
[7]	F. Colasuonno and M. Squassina, Eigenvalues for double phase variational integrals, Ann. Mat. Pura Appl., 2016, 195, 1917–1959. doi: 10.1007/s10231-015-0542-7 CrossRef Google Scholar
[8]	Á. Crespo-Blanco, N. S. Papageorgiou and P. Winkert, Parametric superlinear double phase problems with singular term and critical growth on the boundary, Math. Meth. Appl. Sci., 2022, 45, 2276–2298. doi: 10.1002/mma.7924 CrossRef Google Scholar
[9]	L. Diening, et al., Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math., vol. 2017, Springer, Heidelberg, 2011. Google Scholar
[10]	H. Fröhlich, Theory of electrical breakdown in ionic crystals, Proc. Roy. Soci. of London. Series A. Mathematical and Physical Sciences, 1937, 172, 230–241. Google Scholar
[11]	F. Gao, V. Moroz, M. Yang and S. Zhao, Construction of infinitely many solutions for a critical Choquard equation via local Pohožaev identities, Cal. Var. Par. Diff. Equ., 2022, 61, 1–47. Google Scholar
[12]	B. Ge and P. Pucci, Quasilinear double phase problems in the whole space via perturbation methods, Adv. Diff. Equ., 2022, 27, 1–30. Google Scholar
[13]	B. Ge and W. S. Yuan, Existence of at least two solutions for double phase problem, J. Appl. Anal. & Comp., 2022, 12, 1443–1450. Google Scholar
[14]	P. Harjulehto and P. Hästö, Orlicz Spaces and Generalized Orlicz Spaces, Springer, 2019. Google Scholar
[15]	E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 1977, 57, 93–105. doi: 10.1002/sapm197757293 CrossRef Google Scholar
[16]	E. H. Lieb and M. Loss, "Analysis", Gradute Studies in Mathematics, AMS, Providence, Rhode island, 2001, 2. Google Scholar
[17]	J. Liu, J. F. Liao, H. L. Pan and C. L. Tang, On the Choquard equations under the effect of a general nonlinear term, Top. Meth. Nonlinear Anal., 2022, 1–14. Google Scholar
[18]	W. Liu and G. Dai, Multiplicity results for double phase problems in $\mathbb{R}^{N}$, J. Math. Phy., 2020, 61, 091508. doi: 10.1063/5.0020702 CrossRef $\mathbb{R}^{N}$" target="_blank">Google Scholar
[19]	W. Liu, G. Dai, N. S. Papageorgiou and P. Winkert, Existence of solutions for singular double phase problems via the Nehari manifold method, Anal. Math. Phy., 2022, 12, 75. doi: 10.1007/s13324-022-00686-6 CrossRef Google Scholar
[20]	W. Liu and P. Winkert, Combined effects of singular and superlinear nonlinearities in singular double phase problems in $\mathbb{R}^{N}$, J. Math. Anal. Appl., 2022, 507, 125762. doi: 10.1016/j.jmaa.2021.125762 CrossRef $\mathbb{R}^{N}$" target="_blank">Google Scholar
[21]	L. Maia, B. Pellacci and D. Schiera, Symmetric positive solutions to nonlinear Choquard equations with potentials, Cal. Var. Par. Diff. Equ., 2022, 61, 1–34. Google Scholar
[22]	I. M. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical and Quan. Gravity, 1998, 15, 2733. doi: 10.1088/0264-9381/15/9/019 CrossRef Google Scholar
[23]	V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fix. Point Theory and Appl., 2017, 19, 773–813. doi: 10.1007/s11784-016-0373-1 CrossRef Google Scholar
[24]	J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math., vol. 1034, Springer, 1983, Berlin. Google Scholar
[25]	S. I. Pekar, Untersuchungen uber die Elektronentheorie der Kristalle, Akademie-Verlag, 1954, Berlin. Google Scholar
[26]	Y. Su and S. Liu, Ground state solutions for critical Choquard equation with singular potential: Existence and regularity, J. Fix. Point Theory and Appl., 2023, 25, 23. doi: 10.1007/s11784-022-01032-w CrossRef Google Scholar
[27]	X. Xie, T. Wang and W. Zhang, Existence of solutions for the (p, q)-Laplacian equation with nonlocal Choquard reaction, Appl. Math. Lett., 2023, 135, 108418. doi: 10.1016/j.aml.2022.108418 CrossRef Google Scholar
[28]	S. Yao, J. Sun and T. f. Wu, Positive solutions to a class of Choquard type equations with a competing perturbation, J. Math. Anal. Appl., 2022, 516, 126469. doi: 10.1016/j.jmaa.2022.126469 CrossRef Google Scholar
[29]	W. Zhang, J. Zhang and V. D. Rădulescu, Concentrating solutions for singularly perturbed double phase problems with nonlocal reaction, J. Diff. Equ., 2023, 347, 56–103. doi: 10.1016/j.jde.2022.11.033 CrossRef Google Scholar
[30]	X. Zhang, Y. Meng and X. He, Analysis of a low linear perturbed Choquard equation with critical growth, J. Fix. Point Theory and Appl., 2023, 25, 3. doi: 10.1007/s11784-022-01016-w CrossRef Google Scholar
[31]	J. Zuo, D. Choudhuri and D. D. Repovš, Mixed order elliptic problems driven by a singularity, a Choquard type term and a discontinuous power nonlinearity with critical variable exponents, Fra. Cal. Appl. Anal., 2022, 25, 2532–2553. doi: 10.1007/s13540-022-00105-4 CrossRef Google Scholar