Double phase problem with singularity and homogenous Choquard type term. (English) Zbl 07924040
Summary: 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.
MSC:
| 35A15 | Variational methods applied to PDEs |
| 35J20 | Variational methods for second-order elliptic equations |
| 35J60 | Nonlinear elliptic equations |
| 35B38 | Critical points of functionals in context of PDEs (e.g., energy functionals) |
Keywords:
double phase operator; singular problem; Choquard term; existence of solutions; variational method; Musielak-Orlicz spacesReferences:
| [1] | A. Aberqi, O. Benslimane, M. Elmassoudi and M. A. Ragusa, Nonnegative so-lution of a class of double phase problems with logarithmic nonlinearity, Boun. Val. Prob., 2022, 2022, 1-13. · Zbl 1500.35179 |
| [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. · Zbl 1501.35199 |
| [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. |
| [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. · Zbl 1518.35394 |
| [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. · Zbl 1480.35206 |
| [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. · Zbl 1485.35229 |
| [7] | F. Colasuonno and M. Squassina, Eigenvalues for double phase variational integrals, Ann. Mat. Pura Appl., 2016, 195, 1917-1959. · Zbl 1364.35226 |
| [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. · Zbl 1529.35229 |
| [9] | L. Diening, et al., Lebesgue and Sobolev Spaces with Variable Exponents, Lec-ture Notes in Math., vol. 2017, Springer, Heidelberg, 2011. · Zbl 1222.46002 |
| [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. |
| [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. · Zbl 1501.35192 |
| [12] | B. Ge and P. Pucci, Quasilinear double phase problems in the whole space via perturbation methods, Adv. Diff. Equ., 2022, 27, 1-30. · Zbl 1486.35205 |
| [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. · Zbl 07906560 |
| [14] | P. Harjulehto and P. Hästö, Orlicz Spaces and Generalized Orlicz Spaces, Springer, 2019. · Zbl 1436.46002 |
| [15] | E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math., 1977, 57, 93-105. · Zbl 0369.35022 |
| [16] | E. H. Lieb and M. Loss, “Analysis”, Gradute Studies in Mathematics, AMS, Providence, Rhode island, 2001, 2. · Zbl 0966.26002 |
| [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. |
| [18] | W. Liu and G. Dai, Multiplicity results for double phase problems in R N , J. Math. Phy., 2020, 61, 091508. · Zbl 1454.74028 |
| [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. · Zbl 1496.35185 |
| [20] | W. Liu and P. Winkert, Combined effects of singular and superlinear nonlin-earities in singular double phase problems in R N , J. Math. Anal. Appl., 2022, 507, 125762. · Zbl 1490.35168 |
| [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. · Zbl 1485.35238 |
| [22] | I. M. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical and Quan. Gravity, 1998, 15, 2733. · Zbl 0936.83037 |
| [23] | V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fix. Point Theory and Appl., 2017, 19, 773-813. · Zbl 1360.35252 |
| [24] | J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math., vol. 1034, Springer, 1983, Berlin. · Zbl 0557.46020 |
| [25] | S. I. Pekar, Untersuchungen uber die Elektronentheorie der Kristalle, Akademie-Verlag, 1954, Berlin. · Zbl 0058.45503 |
| [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. · Zbl 1505.35186 |
| [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. · Zbl 1500.35191 |
| [28] | S. Yao, J. Sun and T. f. Wu, Positive solutions to a class of Choquard type equa-tions with a competing perturbation, J. Math. Anal. Appl., 2022, 516, 126469. · Zbl 1497.35261 |
| [29] | W. Zhang, J. Zhang and V. D. Rȃdulescu, Concentrating solutions for sin-gularly perturbed double phase problems with nonlocal reaction, J. Diff. Equ., 2023, 347, 56-103. · Zbl 1505.35024 |
| [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. · Zbl 1505.35187 |
| [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. · Zbl 1503.35281 |
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