Abstract
In this present paper, we investigate a new class of singular double phase p-Laplacian equation problems with a \(\psi \)-Hilfer fractional operator combined from a parametric term. Motivated by the fibering method using the Nehari manifold, we discuss the existence of at least two weak solutions to such problems when the parameter is small enough. Before attacking the main contribution, we discuss some results involving the energy functional and the Nehari manifold.
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Baroni, P., Colombo, M., Mingione, G.: Harnack inequalities for double phase functionals. Nonlinear Anal. 121, 206–222 (2015)
Baroni, P., Colombo, M., Mingione, G.: Non-autonomous functionals, borderline cases and related function classes. St. Petersburg Math. J. 27, 347–379 (2016)
Bahrouni, A., Radulescu, V.D., Winkert, P.: Double phase problems with variable growth and convection for the Baouendi–Grushin operator. Z. Angew. Math. Phys. 71(6), 183, 14 pp (2020)
Colasuonno, F., Squassina, M.: Eigenvalues for double phase variational integrals. Ann. Mat. Pura Appl. 4(6), 195, 1917–1959 (2016)
Colombo, M., Mingione, G.: Bounded minimisers of double phase variational integrals. Arch. Ration. Mech. Anal. 218(1), 219–273 (2015)
Colombo, M., Mingione, G.: Regularity for double phase variational problems. Arch. Ration. Mech. Anal. 215, 443–496 (2015)
Cui, N., Sun, Hong-Rui.: Existence and multiplicity results for double phase problem with nonlinear boundary condition. Nonlinear Anal. Real World Appl. 60, 103307 (2021)
Gasínski, L., Papageorgiou, N.S.: Constant sign and nodal solutions for superlinear double phase problems. Adv. Calc. Var. 14(4), 613–626 (2021)
Ghanmi, A., Zhang, Z.: Nehari manifold and multiplicity results for a class of fractional boundary value problems with \(p\)-Laplacian. Bull. Korean Math. Soc. 56(5), 1297–1314 (2019)
Hewitt, E., Stromberg, K.: Real and Abstract Analysis. Springer, New York (1965)
Jiao, F., Zhou, Y.: Existence of solutions for a class of fractional boundary value problems via critical point theory. Comput. Math. Appl. 62(3), 1181–1199 (2011)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, vol. 204. Elsevier, Amsterdam (2006)
Lei, C.-Y.: Existence and multiplicity of positive solutions for Neumann problems involving singularity and critical growth. J. Math. Anal. Appl. 459(2), 959–979 (2018)
Liu, W., Dai, G.: Existence and multiplicity results for double phase problem. J. Differ. Equ. 265(9), 4311–4334 (2018)
Machado, J.A., Tenreiro: The bouncing ball and the Grünwald–Letnikov definition of fractional operator. Frac. Calc. Appl. Anal. 24(4), 1003–1014 (2021)
Musielak, J.: Orlicz Spaces and Modular Spaces. Springer, Berlin (1983)
Nemati, S., Lima, Pedro M., Torres, Delfim F. M.: A numerical approach for solving fractional optimal control problems using modified hat functions. Commun. Nonlinear Sci. Numer. Simul. 78, 104849 (2019)
Norouzi, F., N’Guérékata, Gaston M.: A study of \(\psi \)-Hilfer fractional differential system with application in financial crisis. Chaos Solitons Fractals 6, 100056 (2021)
Norouzi, F., N’guérékata, G.M.: Existence results to a \(\psi \)-Hilfer neutral fractional evolution equation with infinite delay. Nonautonomous Dyn. Syst. 8(1), 101–124 (2021)
Nyamoradi, N., Tayyebi, E.: Existence of solutions for a class of fractional boundary value equations with impulsive effects via critical point theory. Mediterr. J. Math. 15(3), 1–25 (2018)
Odibat, Z., Erturk, V.S., Kumar, P., Govindaraj, V.: Dynamics of generalized Caputo type delay fractional differential equations using a modified Predictor-Corrector scheme. Phys. Scr. 96(12), 125213 (2021)
Ok, J.: Partial regularity for general systems of double phase type with continuous coefficients. Nonlinear Anal. 177, 673–698 (2018)
Papageorgiou, N.S., Repovs, D.D., Vetro, C.: Positive solutions for singular double phase problems. J. Math. Anal. Appl. 501(1), 123896 (2021)
Ragusa, M.A., Tachikawa, A.: Regularity for minimizers for functionals of double phase with variable exponents. Adv. Nonlinear Anal. 9(1), 710–728 (2020)
Silva, C.J., Torres, D.F.M.: Stability of a fractional HIV/AIDS model. Math. Comput. Simul. 164, 180–190 (2019)
Sousa, J.V.C., dos Santos, N.N.S., da Costa, E., Magna, L.A., de Oliveira, E.C.: A new approach to the validation of an ESR fractional model. Comput. Appl. Math. 40(3), 1–20 (2021)
Sousa, J.V.C., Tavares, L.S., César, E., Torres, L.: A variational approach for a problem involving a \(\psi \)-Hilfer fractional operator. J. Appl. Anal. Comput. 11(3), 1610–1630 (2021)
Sousa, J.V.C., Mouffak, B., N’Guérékata, G.M.: Attractivity for differential equations of fractional order and \(\psi \)-Hilfer type. Frac. Cal. Appl. Anal. 23(4), 1188–1207 (2020)
Sousa, J.V.C., Aurora, M., Pulido, P., de Oliveira, E.C.: Existence and regularity of weak solutions for \(\psi \)-Hilfer fractional boundary value problem. Mediterr. J. Math. 18(4), 1–15 (2021)
Sousa, J.V.C., Tavares, L.S., de Oliveira, E.C.: Existence and Uniqueness of mild and strong for fractional evolution equation. Palestine J. Math. 10(2) (2021)
Sousa, J.V.C., De Oliveira, E.C.: Leibniz type rule: \(\psi \)-Hilfer fractional operator. Commun. Nonlinear Sci. Numer. Simul. 77, 305–311 (2019)
Sousa, J.V.C.: Nehari manifold and bifurcation for a \(\psi \)-Hilfer fractional \(p\)-Laplacian. Math. Meth. Appl. Sci. (2021)
Sousa, J.V.C., Sousa, C.T.L., Pigossi, M., Zuo, J.: Nehari manifold for weighted singular fractional \(\psi \)-Laplace equations. Prepint (2021)
Sousa, J.V.C., De Oliveira, E.C.: On the \(\psi \)-Hilfer fractional operator. Commun. Nonlinear Sci. Numer. Simul. 60, 72–91 (2018)
Sousa, J.V.C., Zuo, J., O’Regan, D.: The Nehari manifold for a \(\psi \)-Hilfer fractional \(p\)-Laplacian. Appl. Anal. 1–31 (2021)
Srivastava, H.M., Sousa, J.V.C.: Multiplicity of solutions for fractional-order differential equations via the \(\kappa (x)\)-Laplacian operator and the genus theory. Fractal Fract. 6(9), 481 (2022)
Suwan, I., Abdo, M., Abdeljawad, T., Mater, M., Boutiara, A., Almalahi, M.: Existence theorems for Psi-fractional hybrid systems with periodic boundary conditions AIMS. AIMS Math. 7(1), 171–186
Vangipuram, L., Vatsala, A.S.: Basic theory of fractional differential equations. Nonlinear Anal. Theory Methods Appl. 69(8), 2677–2682 (2008)
Wulong, L., Dai, G., Papageorgiou, N.S., Winkert, P.: Existence of solutions for singular double phase problems via the Nehari manifold method. arXiv:2101.00593 (2021)
You, Z., Fečkan, M., Wang, J.R.: Relative controllability of fractional delay differential equations via delayed perturbation of Mittag–Leffler functions. J. Comput. Appl. Math. 378, 112939 (2020)
Zhang, Z., Li, J.: Variational approach to solutions for a class of fractional boundary value problem. Electronic J. Quali. Theory Differ. Equ. 2015(11), 1–10 (2015)
Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk SSSR Ser. Mat. 50(4), 675–710 (1986)
Zhao, Y., Tang, L.: Multiplicity results for impulsive fractional differential equations with \(p\)-Laplacian via variational methods. Bound. Value Probl. 2017(1), 1–15 (2017)
Zhikov, S.M., Kozlov, V.V., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994)
Zhikov, V.V.: On Lavrentiev’s phenomenon. Russ. J. Math. Phys. 3, 249–269 (1995)
Zhikov, V.V.: On some variational problems. Russ. J. Math. Phys. 5, 105–116 (1997)
Zhikov, V.V.: On variational problems and nonlinear elliptic equations with nonstandard growth conditions. J. Math. Sci. 173(5), 463–570 (2011)
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All authors’ contributions to this manuscript are the same. All authors read and approved the final manuscript. We are very grateful to the anonymous reviewers for their useful comments that led to improvement of the manuscript.
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Sousa, J.V.d.C., Lima, K.B. & Tavares, L.S. Existence of Solutions for a Singular Double Phase Problem Involving a \(\psi \)-Hilfer Fractional Operator Via Nehari Manifold. Qual. Theory Dyn. Syst. 22, 94 (2023). https://doi.org/10.1007/s12346-023-00794-z
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DOI: https://doi.org/10.1007/s12346-023-00794-z
Keywords
- \(\psi \)-Hilfer fractional operator
- Fractional differential equations
- Double phase operator
- Fibering method
- Multiple solutions
- Nehari manifold
- Singular problem