Existence and multiplicity of solutions for fractional differential equations with p-Laplacian at resonance

Authors

  • Jose Vanterler da C. Sousa Univ. Federal do ABC-UFABC, Santo Andre, SP, Brazil
  • Mariane Pigossi Univ. Federal do Espirito Santo, Vitoria, ES, Brazil
  • Nemat Nyamoradi Razi Univ., Kermanshah, Iran

DOI:

https://doi.org/10.58997/ejde.2024.34

Keywords:

Fractional differential equations; p-Laplacian equation; resonance; existence and multiplicity.

Abstract

In this article, we investigate the existence and multiplicity of solutions for a fractional differential equations with p-Laplacian equation at resonance in the \(\psi\)-fractional space \(H^{\alpha, \beta; \psi}_p\). In addition, we show that the energy functional satisfies the Palais-Smale condition.

For more information see https://ejde.math.txstate.edu/Volumes/2024/34/abstr.html

References

Ahmad, S.; Multiple nontrivial solutions of resonant and non-resonant asymptotically linear problems. Proc. Amer. Math. Soc. 96 (3) (1986), 405-409.

Alsulami, H.; Kirane, M.; Alhodily, S.; Saeed, T.; Nyamoradi, N.; Existence and multiplicity of solutions to fractional p-Laplacian systems with concave-convex nonlinearities. Bull. Math. Sci. 10.01 (2020), 2050007.

Alziary, B.; Fleckinger, J.; Takac, P.; Variational Methods for a Resonant Problem with the p-Laplacian in RN. Electron. J. Differential Equ., 2004 (2004), No. 76, 1-32.

Ambrosetti, A.; Mancini G.; Existence and multiplicity results for nonlinear elliptic problems with linear part at resonance. The case of the simple eigenvalue. J. Differential Equ. 28 (2), (1978), 220-245.

Amiri, S.; Nyamoradi, N.; Behzadi, A.; Ambrosio, V.; Existence and multiplicity of positive solutions to fractional Laplacian systems with combined critical Sobolev terms. Positivity. 25.4 (2021), 1373-1402.

Anane, A.; Tsouli, N.; On a nonresonance condtion between the first and the second eigenvalues for the p-Laplacian, Internat. J. Math. Math. Sci. 26 (2001), 625-634.

Bartolo, P.; Benci, V.; Fortunato, D.; Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity. Nonlinear Analysis: Theory, Methods & Appl. 7.9 (1983), 981.1012.

Bartsch, T.; Li, S.; Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear Analysis: Theory, Methods & Appl. 28.3 (1997), 419.441.

Brown, K. J.; Zhang, Y.; The Nehari manifold for a semilinear elliptic equation with a signchanging weight function. J. Differential Equ., 193.2 (2003), 481.499.

Costa, D. G.; Gon¢¬calves, J. V. A.; Existence and multiplicity results for a class of nonlinear elliptic boundary value problems at resonance. J. Math. Anal. Appl., 84(2) (1981), 328-337.

Chang, X.; Li, Y.; Existence and multiplicity of nontrivial solutions for semilinear elliptic Dirichlet problems across resonance. Topol. Meth. Nonlinear Anal., 36.2 (2010), 285.310.

Chang, K. C.; Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhauser, Boston, 1993.

Diethelm, K.; Ford, N. J.; Analysis of fractional differential equations. J. Math. Anal. Appl. 265.2 (2002), 229.248.

Drabek, P.; Pohozaev, S. I.; Positive solutions for the p-Laplacian: application of the fibrering method. Proc. Royal Soc. Edinburgh Section A: Math. 127.4 (1997), 703.726.

Drabek, P.; Robinson, Stephen B.; Resonance problems for the p-Laplacian. J. Funct. Anal. 169.1 (1999), 189.200.

El Amrouss, A. R.; Nontrivial solutions of semilinear equations at resonance. J. Math. Anal. Appl. 325.1 (2007), 19.35.

Garain, P.; Mukherjee, T.; On a class of weighted p-Laplace equation with singular nonlinearity. Mediterr. J. Math. 17.4 (2020), 1.18.

Gorenflo, R.; F. Mainardi; Fractional calculus. Fractals and fractional calculus in continuum mechanics. Springer, Vienna, 1997. 223-276.

Gupta, C. P.; Solvability of a boundary value problem with the nonlinearity satisfying a sign condition. J. Math. Anal. Appl. 129(2) (1988), 482-492.

Hassani, H.; Machado, J. A. Tenreiro; Mehrabi, S.; An optimization technique for solving a class of nonlinear fractional optimal control problems: Application in cancer treatment. Appl. Math. Modell. 93 (2021), 868.884.

Hilfer, R.; Applications of fractional calculus in physics. World scientific, 2000.

Hilfer, R.; Butzer, P. L.; Westphal, U.; An introduction to fractional calculus. Appl. Fract. Calc. Phys., (2010), 1.85.

Hu, L.; Zhang, S.; Existence of positive solutions to a periodic boundary value problems for nonlinear fractional differential equations at resonance. J. Fract. Calc. Appl. 8.2 (2017), 19.31.

Hu, L.; On the existence of positive solutions for fractional differential inclusions at resonance. Springer Plus 2016 (2016), 1-12.

Imaga, O. F.; Iyase, S. A.; On a fractional-order p-Laplacian boundary value problem at resonance on the half-line with two dimensional kernel. Adv. Difference Equ. 2021.1 (2021), 1.14.

Jiang, W.; Qiu, J.; Yang, C.; The existence of positive solutions for p-Laplacian boundary value problems at resonance. Boundary Value Problems. 2016.1 (2016), 1.9.

Jiang, W; Solvability of fractional differential equations with p-Laplacian at resonance. Appl. Math. Comput. 260 (2015), 48.56.

Jiang, W.; Qiu, J., Yang, C.; The existence of solutions for fractional differential equations with p-Laplacian at resonance. Chaos: An Inter. J. Nonlinear Sci. 27.3 (2017), 032102.

Jiu, Q.; Su, J.; Existence and multiplicity results for Dirichlet problems with p-Laplacian. J. Math. Anal. Appl. 281.2 (2003), 587.601.

Kilbas, A. A.; Srivastava, Hari M; Trujillo, Juan J.; Theory and applications of fractional differential equations. Vol. 204. Elsevier, 2006.

Kosmatov, N. A; Boundary value problem of fractional order at resonance, Electron. J. Differ. Equ. 2010 (2010), No. 35, 1.10.

Landesman, E. M.; Lazer, A. C.; Nonlinear perturbations of linear elliptic boundary value problems at resonance. J. Math. Mech. 19 (1970), 609-623.

Liu, J., S; Wang, Zhang, J.; Multiple solutions for boundary value problems of second-order difference equations with resonance. J. Math. Anal. Appl. 374.1 (2011), 187.196.

Meral, F. C.; Royston, T. J., Magin, R.; Fractional calculus in viscoelasticity: an experimental study. Commun. Nonlinear Sci. Numer. Simul. 15 (2010), 939.945.

Nigmatullin, R.; Omay, T.; Baleanu, D.; On fractional filtering versus conventional filtering in economics. Commun. Nonlinear Sci. Numer. Simul. 15 (2010), 979.986.

Nikan, O.; Avazzadeh, Z.; Machado, J. A. Tenreiro; Numerical approximation of the nonlinear time-fractional telegraph equation arising in neutron transport. Commun. Nonlinear Sci. Numer. Simul. 99 (2021), 105755.

Nyamoradi, N.; Tersian, S.; Existence of solutions for nonlinear fractional order p-Laplacian differential equations via critical point theory. Frac. Cal. Appl. Anal. 22.4 (2019), 945-967.

Oldham, K.; Fractional differential equations in electrochemistry. Adv. Eng. Softw. 41 (2010), 9.12.

Orsingher, E.; Beghin, L.; Time-fractional telegraph equations and telegraph processes with brownian time. Probab. Theory. Related. Fields. 128 (2004), 141.160.

Sousa, J. Vanterler da C.; Zuo, J.; O¡¯Regan, Donal; The Nehari manifold for a ¥÷-Hilfer fractional p-Laplacian. Applicable Anal. (2021), 1-31.

Sousa, J. Vanterler da C.; Tavares, Leandro S.; Torres, Cesar E.; A variational approach for a problem involving a ¥÷-Hilfer fractional operator. J. Appl. Anal. Comput. 11(3) (2021), 1610-1630.

Sousa, J. Vanterler da C; Nehari manifold and bifurcation for a ¥÷-Hilfer fractional p- Laplacian. Math. Meth. Appl. Sci. 44.11 (2021), 9616-9628.

Sousa, J. Vanterler da C.; Oliveira, E. Capelas de; On the ¥÷-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 60 (2018), 72-91.

Sousa, J. Vanterler da C.; Oliveira, E. Capelas de; Leibniz type rule: ¥÷-Hilfer fractional operator. Commun. Nonlinear Sci. Numer. Simul. 77 (2019), 305.311.

Sousa, J. Vanterler da C.; Santos, Magnun N. N. dos; Costa, E. da; Magna, L. A., Oliveira, E. Capelas de; A new approach to the validation of an ESR fractional model. Comput. Appl. Math. 40.3 (2021), 1.20.

Sousa, J. Vanterler da C.; Oliveira, E. Capelas de; Magna, L. A.; Fractional calculus and the ESR test. AIMS Mathematics, 2.4 (2017), 692.705.

Sun, M.-Zheng; Existence results for the p-Laplacian equation with resonance at the first two eigenvalues. Topol. Meth. Nonlinear Anal. 39.1 (2012), 93.105.

Sun, M.; Multiplicity of solutions for a class of the quasilinear elliptic equations at resonance. J. Math. Anal. Appl. 386.2 (2012), 661.668.

Xue, T.; Fan, X., Xu, J.; Existence of Positive Solutions for a Kind of Fractional Multi-point Boundary Value Problems at Resonance. Inter. J. Appl. Math. 49.3 (2019), 1.8.

Wu, Y.; Liu, W.; Positive solutions for a class of fractional differential equations at resonance. Adv. Differ. Equ. 2015 (2015), 1.11.

Zheng, Y.; Z. B. Fang; Critical curves for a fast diffusive p-Laplacian equation with nonlocal source. Applicable Anal. 101.9 (2022), 3389-3409.

Zhou, Y.; Wang, J.; Zhang, L.; Basic theory of fractional differential equations. World scientific, 2016.

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2024-04-26

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Existence and multiplicity of solutions for fractional differential equations with p-Laplacian at resonance. (2024). Electronic Journal of Differential Equations, 2024(01-??), No. 34, 1-17. https://doi.org/10.58997/ejde.2024.34