| Citation: |
Yi Wang, Lixin Tian. SOLVABILITY FOR COUPLED IMPULSIVE FRACTIONAL PROBLEMS OF THE KIRCHHOFF TYPE WITH $P\&Q$-LAPLACIAN[J]. Journal of Applied Analysis & Computation, 2024, 14(6): 3099-3133. doi: 10.11948/20230465
|
SOLVABILITY FOR COUPLED IMPULSIVE FRACTIONAL PROBLEMS OF THE KIRCHHOFF TYPE WITH $P\&Q$-LAPLACIAN
-
Abstract
In this paper, we investigate the existence and multiplicity of nontrivial solutions for the $p\&q$-Laplacian Kirchhoff impulsive fractional differential equations through variational methods. By utilizing the Nehari manifold and fibering maps, we establish the existence of at least one nontrivial solution to such equations for any $(\lambda,\mu)\in\Theta_{\ast}$. Furthermore, using the idea of truncation arguments and Krasnoselskii genus theory, we demonstrate the existence of infinitely many nontrivial solutions for the equation when Kirchhoff functions $M_{1}$ and $M_{2}$ are degenerate considering any $(\lambda,\mu)\in\Theta_{\ast\ast}$.
-
-
References
[1] G. Afrouzi, S. Heidarkhani and S. Moradi, Multiple solutions for a Kirchhoff-type second-order impulsive differential equation on the half-line, Quaest. Math., 2022, 45(1), 109–141. doi: 10.2989/16073606.2020.1850537 [2] R. Agarwal, M. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta. Appl. Math., 2010, 109, 973–1033. doi: 10.1007/s10440-008-9356-6 [3] K. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differ. Equ., 2003, 193(2), 481–499. doi: 10.1016/S0022-0396(03)00121-9 [4] M. Chaves, G. Ercole and O. Miyagaki, Existence of a nontrivial solution for the $(p, q)$-Laplacian in $R.{N}$ without the Ambrosetti-Rabinowitz condition, Nonlinear Anal., 2015, 114, 133–141. doi: 10.1016/j.na.2014.11.010 CrossRef $(p, q)$-Laplacian in
$R.{N}$ [5] T. Chen and W. Liu, Ground state solutions of Kirchhoff-type fractional Dirichlet problem with $p$-Laplacian, Adv. Differ. Equ., 2018, 2018, 1–9. doi: 10.1186/s13662-017-1452-3 [6] T. Chen, W. Liu and H. Jin, Nontrivial solutions of the Kirchhoff-type fractional $p$-Laplacian Dirichlet problem, J. Funct. Spaces, 2020, 2020, 1–8. $p$-Laplacian Dirichlet problem" target="_blank">Google Scholar
[7] K. Diethelm, The Analysis of Fractional Differential Equation, Spring, New York, 2010. [8] C. Farkas, A. Fiscella and P. Winkert, On a class of critical double phase problems, J. Math. Anal. Appl., 2022, 515(2), 1–16. [9] A. Fiscella and P. Mishra, The Nehari manifold for fractional Kirchhoff problems involving singular and critical terms, Nonlinear Anal., 2019, 186, 6–32. doi: 10.1016/j.na.2018.09.006 [10] A. Fiscella and P. Pucci, Degenerate Kirchhoff $(p, q)$-fractional systems with critical nonlinearities, Fract. Calc. Appl. Anal., 2020, 23(3), 723–752. doi: 10.1515/fca-2020-0036 CrossRef $(p, q)$-fractional systems with critical nonlinearities" target="_blank">Google Scholar
[11] J. Graef, S. Heidarkhani, L. Kong and S. Moradi, Three solutions for impulsive fractional boundary value problems with $p$-Laplacian, Bull. Iran Math. Soc., 2022, 48(4), 1413–1433. doi: 10.1007/s41980-021-00589-5 [12] S. Heidarkhani and A. Salari, Nontrivial solutions for impulsive fractional differential systems through variational methods, Math. Methods. Appl. Sci., 2020, 43(10), 6529–6541. doi: 10.1002/mma.6396 [13] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. [14] F. Jiao and Y. Zhou, Existence results for fractional boundary value problem via critical point theory, Int. J. Bifurcation Chaos, 2012, 22(4), 1–17. [15] M. Kratou, Ground state solutions of $p$-Laplacian singular Kirchhoff problem involving a Riemann-Liouville fractional derivative, Filomat, 2019, 33(7), 2073–2088. doi: 10.2298/FIL1907073K CrossRef $p$-Laplacian singular Kirchhoff problem involving a Riemann-Liouville fractional derivative" target="_blank">Google Scholar
[16] D. Li, F. Chen and Y. An, Positive solutions for a $p$-Laplacian type system of impulsive fractional boundary value problem, J. Appl. Anal. Comput., 2020, 10(2), 740–759. [17] D. Li, F. Chen and Y. An, The existence of solutions for an impulsive fractional coupled system of $(p, q)$-Laplacian type without the Ambrosetti-Rabinowitz condition, Math. Meth. Appl. Sci., 2019, 42(5), 1449–1464. doi: 10.1002/mma.5435 CrossRef $(p, q)$-Laplacian type without the Ambrosetti-Rabinowitz condition" target="_blank">Google Scholar
[18] D. Li, F. Chen and Y. An, Variational formulation for nonlinear impulsive fractional differential equations with $(p, q)$-Laplacian operator, Math. Meth. Appl. Sci., 2022, 45(1), 515–531. doi: 10.1002/mma.7791 CrossRef $(p, q)$-Laplacian operator" target="_blank">Google Scholar
[19] N. Nyamoradi and S. Tersian, Existence of solutions for nonlinear fractional order $p$-Laplacian differential equations via critical point theory, Fract. Calc. Appl. Anal., 2019, 22(4), 945–967. doi: 10.1515/fca-2019-0051 CrossRef $p$-Laplacian differential equations via critical point theory" target="_blank">Google Scholar
[20] K. Perera, M. Squassina and Y. Yang, Bifurcation and multiplicity results for critical fractional $p$-Laplacian problems, Math. Nachr., 2016, 289(2–3), 332–342. doi: 10.1002/mana.201400259 CrossRef $p$-Laplacian problems" target="_blank">Google Scholar
[21] P. Pucci and L. Temperini, Existence for fractional $(p, q)$ systems with critical and Hardy terms in $R.{N}$, Nonlinear Anal., 2021, 211, 1–33. $(p, q)$ systems with critical and Hardy terms in
$R.{N}$ [22] R. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, American Mathematical Society Providence, Rhode Island, 1986. [23] R. Stegliński, Infinitely many solutions for double phase problem with unbounded potential in $R.{N}$, Nonlinear Anal., 2022, 214, 1–20. [24] G. Tarantello, On nonhomogeneous elliptic involving critical Sobolev exponent, Ann. Inst. Henri Poincar-Analyse non linéaire, 1992, 9(3), 281–304. doi: 10.1016/s0294-1449(16)30238-4 [25] L. Wang, K. Xie and B. Zhang, Existence and multiplicity of solutions for critical Kirchhoff-type $p$-Laplacian problems, J. Math. Anal. Appl., 2018, 458(1), 361–378. doi: 10.1016/j.jmaa.2017.09.008 CrossRef $p$-Laplacian problems" target="_blank">Google Scholar
[26] M. Xiang, D. Hu and D. Yang, Least energy solutions for fractional Kirchhoff problems with logarithmic nonlinearity, Nonlinear Anal., 2020, 198, 1–20. [27] M. Xiang, B. Zhang and D. Repovš, Existence and multiplicity of solutions for fractional Schrödinger-Kirchhoff equations with Trudinger-Moser nonlinearity, Nonlinear Anal., 2019, 186, 74–98. doi: 10.1016/j.na.2018.11.008 [28] W. Xie and H. Chen, Multiple positive solutions for the critical Kirchhoff type problems involving sign-changing weight functions, J. Math. Anal. Appl., 2019, 479(1), 135–161. doi: 10.1016/j.jmaa.2019.06.020 [29] Y. Zhao and L. Tang, Multiplicity results for impulsive fractional differential equations with $p$-Laplacian via variational methods, Bound. Value. Probl., 2017, 2017, 1–15. doi: 10.1186/s13661-016-0733-1 CrossRef $p$-Laplacian via variational methods" target="_blank">Google Scholar
[30] J. Zhou, Y. Liu, Y. Wang and J. Suo, Solvability of nonlinear impulsive generalized fractional differential equations with ($p, q$)-Laplacian operator via critical point theory, Fractal Fract., 2022, 6(12), 1–24. $p, q$)-Laplacian operator via critical point theory" target="_blank">Google Scholar
-
-
Export File
Citation
Yi Wang, Lixin Tian. SOLVABILITY FOR COUPLED IMPULSIVE FRACTIONAL PROBLEMS OF THE KIRCHHOFF TYPE WITH $P\&Q$-LAPLACIAN[J]. Journal of Applied Analysis & Computation, 2024, 14(6): 3099-3133. doi: 10.11948/20230465
DownLoad: