Three weak solutions for a class of quasilinear Choquard equations involving the fractional \(p(x, .)\)-Laplacian operator with weight. (English) Zbl 07905294
Summary: In this paper, we establish the existence of at least three weak solutions to a problem involving the fractional \(p(x, .)\)-Laplacian operator with weight. Our method used for obtaining the existence of three solutions for a class of Choquard equations is based on the variational method concerned a type of version of Ricceri.
MSC:
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 35A15 | Variational methods applied to PDEs |
Keywords:
quasilinear Choquard equations; fractional \(p(x, .)\)-Laplacian with weight; variational methods; nonlinear elliptic equationsReferences:
| [1] | C.O. Alves, V. Rȃdulescu, L. S. Tavares., Generalized Choquard equations driven by nonhomoge-neous operators, Mediterr J Math. 16-20, (2019). · Zbl 1411.35124 |
| [2] | C.O. Alves, Tavares LS., A Hardy-Littlewood-Sobolev-type inequality for variable exponents and applications to Quasi-linear Choquard equations involving variable exponent, Meditterr J Math. 16(2), 55(2019). · Zbl 1414.35009 |
| [3] | E. Azroul, A. Benkirane, M. Shimi., Eigenvalue problems involving the fractional p(x)-Laplacian operator, Adv. Oper. Theory4, 539-555, (2019). · Zbl 1406.35456 |
| [4] | E. Azroul, A. Benkirane, M. Shimi and M. Srati., Three solutions for fractional p(x, .)-Laplacian Dirichlet Problems with weight. J. Nonlinear Funct. Anal, (2020). |
| [5] | I. Aydin., Weighted Variable Sobolev Spaces and Capacity, J. Funct. Spaces Appl. Article ID 132690, (2012). · Zbl 1241.46019 |
| [6] | A. Bahrouni, V. Rȃdulescu., On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent, Discrete Contin. Dyn. Syst. 11, 379-389, (2018). · Zbl 1374.35158 |
| [7] | R. Biswas and S. Tiwari., Variable order nonlocal Choquard problem with variable exponents, Complex Variables and Elliptic Equations.1747-6933(Print)1747-6941(Online), (2020). |
| [8] | Y. Chen, S. Levine, M. Rao., Variable exponent, linear growth functionals in image processing, SIAMJ. Appl. Math. 66, 1383-1406, (2006). · Zbl 1102.49010 |
| [9] | N. T Chung and H. Q. Toan., On a class of fractional Laplacian problems with variable exponents and indefinite weights, Collectanea Math. 71, 223-237, (2019). · Zbl 1437.35291 |
| [10] | D. Cruz-Uribe, L. Diening, P. Hästö., The maximal operator on weighted variable Lebesgue spaces, Frac. Calc. Appl. Anal. 14, 361-374, (2011). · Zbl 1273.42018 |
| [11] | S. G. Deng., A local mountain pass theorem and applications to a double perturbed p(x)-laplacian equations, J. Applied Mathematics and computation 211, 234-241, (2009). · Zbl 1173.35045 |
| [12] | X. L. Fan, S.G. Deng., Remarks on Ricceri’s variational principle and applications to the p(x)-Laplacian equations, J. Nonlinear Anal 67, 3064-3075, (2007). · Zbl 1134.35035 |
| [13] | X. L. Fan, D. Zhao., On the spaces L p(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl. 263, 424-446, (2001). · Zbl 1028.46041 |
| [14] | A. El Amrouss and A. Ourraoui., Existence of solutions for a boundary problem involving p(x)-biharmonic operator, Bol. Soc. Paran. Mat. (3s.) v. 31 1, 179-192, (2013). · Zbl 1413.35193 |
| [15] | U. Kaufmann,J. D. Rossi,R. Vidal., Fractional Sobolev spaces with variable exponents and fractional p(x)-Laplacians, Electron. J. Qual. Theory Differ Equ.76; 1-10, (2017) · Zbl 1413.46035 |
| [16] | O. Kovácik, J. Rákosn ík., On spaces L p(x) (Ω) and W m,p(x) (Ω), Czechoslovak Math. J. 41, 592-618, (1991). · Zbl 0784.46029 |
| [17] | M. Råužička., Electrorheological fluids: modeling and mathematical theory, In: Lecture Notes in Mathematics, Vol.1748.Springer, Berlin, 2000. · Zbl 0962.76001 |
| [18] | V. V. Zhikov., Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv. 9, 33-66, (1987). · Zbl 0599.49031 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
