Existence Results for Double Phase Problem Involving Fractional Operators and Singular Nonlinearity


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Authors: R. CHAMMEM, A. GHANMI AND M. MECHERGUI

DOI: 10.46793/KgJMat2609.1377C

Abstract:

The purpose of this paper is to extend some existing results related to the fractional Laplacian operator to a more general fractional operator. More precisely, we combine the variational method with the min-max problem, to study the existence of solutions for some singular problems with double-phase nonlocal operators. To validate our main results an illustrative example is presented.



Keywords:

Elliptic equation, variational method, min-max, nonlocal operator, singular problem.



References:

[1]   R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

[2]   A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122(2) (1994), 519–543. https://doi.org/10.1006/jfan.1994.1078

[3]   G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in N, J. Differ. Equ. 255(8) (2013), 2340–2362. https://doi.org/10.1016/j.jde.2013.06.016

[4]   A. Bahrouni, V. D. Rǎdulescu and D. D. Repovš, Double phase transonic flow problems with variable growth, nonlinear patterns and stationary waves, Nonlinearity 32(7) (2019), 2481–2495. https://doi.org/10.1088/1361-6544/ab0b03

[5]   B. Barrios, E. Colorado, A. De Pablo and U. Sanchez, On some critical problems for the fractional Laplacian operator, J. Diffe. Equ. 252 (2012), 6133–6162. https://doi.org/10.1016/j.jde.2012.02.023

[6]   H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Masson, Paris, 1983.

[7]   X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224(5) (2010), 2052–2093. https://doi.org/10.1016/j.aim.2010.01.025

[8]   L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integrodifferential equations, Commun. Pure Appl. Math. 62(5) (2009), 597–638. https://doi.org/10.1002/cpa.20274

[9]   A. Capella, Solutions of a pure critical exponent problem involving the half-Laplacian in annular-shaped domains, Commun. Pure Appl. Anal. 10(6) (2011), 1645–1662. https://doi.org/10.3934/cpaa.2011.10.1645

[10]   R. Chammem, A. Ghanmi and M. Mechergui, Combined effects in nonlinear elliptic equations involving fractional operators, J. Pseudo-Differ. Oper. Appl. 14(3) (2023), Article ID 35. https://doi.org/10.1007/s11868-023-00530-w

[11]   R. Chammem, A. Ghanmi and M. Mechergui, Existence of solutions for a singular double phase Kirchhoff type problems involving the fractional q(x,)-Laplacian operator, Complex Anal. Oper. Theory 18, (2024), Article ID 25. https://doi.org/10.1007/s11785-023-01470-5

[12]   R. Chammem, A. Ghanmi and A. Sahbani, Existence of solution for a singular fractional Laplacian problem with variable exponents and indefinite weights, Complex Var. Elliptic Equ. 66(8) (2020), 1320–1332. https://doi.org/10.1080/17476933.2020.1756270

[13]   M. Chhetri, P. Girg and E. Hollifield, Existence of positive solutions for fractional Laplacian equations: Theory and numerical experiments, Elect. J. Differ. Equ. 2020(81) (2020), 81–31. https://doi.org/10.58997/ejde.2020.81

[14]   A. Crespo-Blanco, L. Gasiňski, P. Harjulejto and P. Winkert, A new class of double phase variable exponent problems: Existence and uniqueness, J. Differ. Equ. 323 (2022), 182–228. https://doi.org/10.1016/j.jde.2022.03.029

[15]   H. Dong and D. Kim, On Lp-estimates for a class of non-local elliptic equations, J. Funct. Anal. 262(3) (2012), 1166–1199. https://doi.org/10.1016/j.jfa.2011.11.002

[16]   A. Fiscella, A fractional Kirchhoff problem involving a singular term and a critical nonlinearity, Adv. Nonlinear Anal. 8(1) (2019), 645–660. https://doi.org/10.1515/anona-2017-0075

[17]   A. Fiscella and P. K. Mishra, The Nehari manifold for fractional Kirchhoff problems involving singular and critical terms, Nonlinear Anal. 186(2019), 6–32. https://doi.org/10.1016/j.na.2018.09.006

[18]   L. Gasiňski and N. S. Papageorgiou, Double phase logistic equations with superdiffusive reaction, Nonlinear Anal. Real World Appl. 70 (2023), Article ID 103782. https://doi.org/10.1016/j.nonrwa.2022.103782

[19]   A. Ghanmi, Existence of nonnegative solutions for a class of fractional p-Laplacian problems, Nonlinear Stud. 22(3) (2015), 373–379.

[20]   A. Ghanmi and K. Saoudi, A multiplicity results for a singular problem involving the fractional p-Laplacian operator, Complex Var. Elliptic Equ. 61(9) (2016), 1199–1216. https://doi.org/10.1080/17476933.2016.1154548

[21]   A. Ghanmi and K. Saoudi, The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator, Fract. Differ. Calc. 6(2) (2016), 201–217. http://dx.doi.org/10.7153/fdc-06-13

[22]   A. Ghanmi and K. Saoudi, A multiplicity results for a singular equation involving the p(x)-Laplace operator, Complex Var. Elliptic Equ. 62(5) (2017), 695–725. https://doi.org/10.1080/17476933.2016.1238466

[23]   A. Ghanmi, Multiplicity of nontrivial solutions of a class of fractional p-Laplacian oroblem, Z. Anal. Anwend. 34(3) (2015), 309–319. https://doi.org/10.4171/ZAA/1541

[24]   J. Giacomoni, D. Kumar and K. Sreenadh, Interior and boundary regularity results for strongly nonhomogeneous p,q-fractional problems, Adv. Calc. Var. 16(2) (2023), 467–501. https://doi.org/10.1515/acv-2021-0040

[25]   C. Huyuan, B. Mousomi and H. Hichem, On the bounds of the sum of eigenvalues for a Dirichlet problem involving mixed fractional Laplacians, J. Diff. Equ. 317 (2022), 31 pages. https://doi.org/10.1016/j.jde.2022.02.004

[26]   O. Kavian, Introduction à la Théorie des Points Critiques et Applications aux Problèmes Elliptiques, Springer-Verlag, Paris, New York, 1993.

[27]   J. F. Liao, X. F. Ke, C. Y. Lei and C. L. Tang, A uniqueness result for Kirchhoff type problems with singularity, Appl. Math. Lett. 59 (2016), 24–30. https://doi.org/10.1016/j.aml.2016.03.001

[28]   E. D. Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136(5) (2012), 521–573. https://doi.org/10.1016/j.bulsci.2011.12.004

[29]   N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Existence and multiplicity of solutions for double phase Robin problems, Bull. Lond. Math. Soc. 52(3) (2020), 546–560. https://doi.org/10.1112/blms.12347

[30]   K. Perera and M. Squassina, Existence results for double-phase problems via Morse theory, Commun. Contemp. Math. 20(2) (2018), Article ID 17500023, https://doi.org/10.1142/S0219199717500237

[31]   K. Saoudi, A. Ghanmi and S. Horrigue, Multiplicity of solutions for elliptic equations involving fractional operator and sign-changing nonlinearity, J. Pseudo-Differ. Oper. Appl. 11(4) (2020), 1743–1756. https://doi.org/10.1007/s11868-020-00357-9

[32]   R. Servadei, The Yamabe equation in a non-local setting, Adv. Nonlinear Anal. 2(3) (2013), 235–270. https://doi.org/10.1515/anona-2013-0008

[33]   R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389(2), 887–898. https://doi.org/10.1016/j.jmaa.2011.12.032

[34]   R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. 33(5) (2013), 2105–2137. https://doi.org/10.3934/dcds.2013.33.2105

[35]   K. Teng, Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators, Nonlinear Anal. Real World Appl. 14(1) (2013), 867–874. https://doi.org/10.1016/j.nonrwa.2012.08.008

[36]   K. Teng, Multiplicity results for hemivariational inequalities driven by nonlocal elliptic operators, J. Math. Anal. Appl. 396 (2012) 386–395. https://doi.org/10.1016/j.jmaa.2012.06.04

[37]   L. Wang, K. Cheng and B. Zhang, A uniqueness result for strong singular Kirchhoff-type fractional Laplacian problems, Appl. Math. Opt. 83 (2021), 1859–1875. https://doi.org/10.1007/s00245-019-09612-y

[38]   P. Wu, Y, Huang and Y. Zhou, Existence and regularity of solutions for a class of fractional Laplacian problems, J. Differ. Equ. 318(5) (2022), 480–501. https://doi.org/10.1016/j.jde.2022.02.041

[39]   V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR-Izv. 29(1) (1987), 33–66. https://doi.org/10.1070/IM1987v029n01ABEH000958

[40]   S. Leonardi and N. S. Papageorgiou, Anisotropic Dirichlet double phase problems with competing nonlinearities, Rev. Mat. Complut. 36(2) (2023), 469–490. https://doi.org/10.1007/s13163-022-00432-3