Research Article
BibTex RIS Cite

Existence and multiplicity of weak solutions for gradient-type systems with oscillatory nonlinearities on the Sierpiński gasket

Year 2019, Volume: 48 Issue: 5, 1461 - 1478, 08.10.2019

Haiffa Muhsan B. Alrikabi Ghasem A. Afrouzi , Mohsen Alimohammady

Abstract

In this paper, we establish the existence and multiplicity results of solutions for parametric quasi-linear systems of the gradient-type on the Sierpiński gasket is proved. Our technical approach is based on variational methods and critical points theory and on certain analytic and geometrical properties of the Sierpiński fractal. Indeed, using a consequence of the local minimum theorem due to Bonanno, the Palais-Smale condition cut off upper at $r$, and the Palais-Smale condition for the Euler functional we investigate the existence of one and two solutions for our problem under algebraic conditions on the nonlinear part. Moreover by applying a different three critical point theorem due to Bonanno and Marano we guarantee the existence of third solution for our problem.

Keywords

gradient-type systems Sierpiński gasket nonlinear elliptic equation Palais-Smale condition

References

  • [1] S. Alexander, Some properties of the spectrum of the Sierpiński gasket in a magnetic field, Phys. Rev. B, 29, 5504–5508, 1984.
  • [2] M.T. Barlow and J. Kigami, Localized eigenfunctions of the Laplacian on p.c.f. self- similar sets, J. London Math. Soc. 56 (2), 320–332, 1997.
  • [3] T. Bartsch and D.G. de Figueiredo, Infinitely many solutions of nonlinear elliptic systems, Progr. Nonlinear Diff. Equ. Appl. 35, 51–67, 1999.
  • [4] A. Bensedik and M. Bouchekif, On certain nonlinear elliptic systems with indefinite terms, Electron. J. Differ. Equ. 83, 1–16, 2002.
  • [5] L. Boccardo and D.G. de Figueiredo, Some remarks on a system of quasilinear elliptic equations, Nonlinear Differ. Equ. Appl. 9, 309–323, 2002.
  • [6] B. Bockelman and R.S. Strichartz, Partial differential equations on products of Sierpiński gaskets, Indiana Univ. Math. J. 56, 1361–1375, 2007.
  • [7] M. Bohner, S. Heidarkhani, A. Salari and G. Caristi, Existence of three solutions for impulsive multi-point boundary value problems, Opuscula Math. 37 (3), 353–379, 2017.
  • [8] G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal. TMA, 75, 2992–3007, 2012.
  • [9] G. Bonanno, Relations between the mountain pass theorem and local minima, Adv. Nonlinear Anal. 1, 205–220, 2012.
  • [10] G. Bonanno and A. Chinnì, Existence and multiplicity of weak solutions for elliptic Dirichlet problems with variable exponent, J. Math. Anal. Appl. 418, 812–827, 2014.
  • [11] G. Bonanno and S.A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal. 89, 1–10, 2010.
  • [12] G. Bonanno, G. Molica Bisci and D. O’Regan, Infinitely many weak solutions for a class of quasilinear elliptic systems, Math. Comput. Modelling, 52, 152–160, 2010.
  • [13] G. Bonanno, G. Molica Bisci and V.D. Rădulescu, Infinitely many solutions for a class of nonlinear elliptic problems on fractals, C. R. Math. Acad. Sci. Paris, 350 (3-4), 187–191, 2012.
  • [14] G. Bonanno, G. Molica Bisci and V.D. Rădulescu, Variational analysis for a nonlinear elliptic problem on the Sierpiński Gasket, ESAIM Control Optim. Calc. Var. 18 (4), 941–953, 2012.
  • [15] G. Bonanno, G. Molica Bisci and V.D. Rădulescu, Qualitative Analysis of Gradient- Type Systems with Oscillatory Nonlinearities on the Sierpiński Gasket, Chin. Ann. Math. 34 (3), 381–398, 2013.
  • [16] Y. Bozhkova and E. Mitidieri, Existence of multiple solutions for quasilinear systems via fibering method, J. Differential Equations, 190, 239–267, 2003.
  • [17] B.E. Breckner, V.D. Rădulescu and C. Varga, Infinitely many solutions for the Dirich- let problem on the Sierpiński gasket, Anal. Appl. 9 (3), 235–248, 2011.
  • [18] B.E. Breckner, D. Repovš and C. Varga, On the existence of three solutions for the Dirichlet problem on the Sierpiński gasket, Nonlinear Anal. 73, 2980–2990, 2010.
  • [19] Ph. Clément, D.G. de Figueiredo and E. Mitidieri, Positive solutions of semilinear elliptic systems, Comm. Partial Differential Equations, 17, 923–940, 1992.
  • [20] D.G. de Figueiredo, Semilinear elliptic systems: a survey of superlinear problems, Resenhas IME-USP, 2, 373–391, 1996.
  • [21] K.J. Falconer, Semilinear PDEs on self-similar fractals, Commun. Math. Phys. 206, 235–245, 1999.
  • [22] K.J. Falconer and J. Hu, Nonlinear elliptical equations on the Sierpiński gasket, J. Math. Anal. Appl. 240, 552–573, 1999.
  • [23] M. Ferrara, G. Molica Bisci and D. Repovs, Existence results for nonlinear elliptic problems on fractal domains, Adv. Nonlinear Anal. 5 (1), 75–84, 2016.
  • [24] M. Fukushima and T. Shima, On a spectral analysis for the Sierpiński gasket, Poten- tial Anal. 1, 1–35, 1992.
  • [25] M. Ghergu and V. Rˇadulescu, Nonlinear PDEs. Mathematical models in biology, chemistry and population genetics, Springer Monographs in Mathematics, Springer, Heidelberg, 2012.
  • [26] S. Goldstein, Random walks and diffusions on fractals, Percolation Theory and Er- godic Theory of Infinite Particle Systems, H. Kesten (ed.), 121–129, IMA Math. Appl., 8, Springer-Verlag, New York, 1987.
  • [27] J.R. Graef, S. Heidarkhani and L. Kong, Multiple solutions for systems of multi-point boundary value problems, Opuscula Math. 33 (2), 293–306, 2013.
  • [28] J.R. Graef, S. Heidarkhani and L. Kong, Infinitely Many Solutions for Systems of Sturm-Liouville Boundary Value Problems, Results. Math. 66, 327–341, 2014.
  • [29] J.R. Graef, S. Heidarkhani and L. Kong, Nontrivial solutions for systems of Sturm- Liouville boundary value problems, Differ. Equ. Appl. 6 (2), 255–265, 2014.
  • [30] J.R. Graef, S. Heidarkhani and L. Kong, Multiple solutions for systems of Sturm- Liouville boundary value problems, Mediterr. J. Math. 13, 1625–1640, 2016.
  • [31] J.R. Graef, S. Heidarkhani and L. Kong, Multiple periodic solutions for perturbed second-order impulsive Hamiltonian systems, Int. J. Pure Appl. Math. 109 (1), 85– 104, 2016.
  • [32] D.D. Hai and R. Shivaji, An existence result on positive solutions for a class of p- Laplacian systems, Nonlinear Anal., 56, 1007–1010, 2004.
  • [33] S. Heidarkhani, Non-trivial solutions for two-point boundary-value problems of fourth- order Sturm-Liouville type equations, Electron. J. Diff. Equ. 2012 (27), 1–9, 2012.
  • [34] S. Heidarkhani, A. Salari and G. Caristi, Kirchhoff-type second-order impulsive dif- ferential equations on the half-line, preprint.
  • [35] J. Hu, Multiple solutions for a class of nonlinear elliptic equations on the Sierpiński gasket, Sci. China Ser. A, 47, 772–786, 2004.
  • [36] C. Hua and H. Zhenya, Semilinear elliptic equations on fractal sets, Acta Math. Sci. Ser. B, 29 (2), 232–242, 2009.
  • [37] J.E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30, 713–747, 1981.
  • [38] S. Kusuoka, A diffusion process on a fractal, Probabilistic Methods in Mathematical Physics, Katata/Kyoto, 1985, 251-274, Academic Press, Boston, 1987.
  • [39] S.M. Kozlov, Harmonization and homogenization on fractals, Communications Math. Phys. 153, 339–357, 1993.
  • [40] G. Molica Bisci and V. Rˇadulescu, A characterization for elliptic problems on fractal sets, Proc. Amer. Math. Soc. 143 (7), 2959–2968, 2015.
  • [41] G. Molica Bisci and V. Rˇadulescu, A sharp eigenvalue theorem for fractional elliptic equations, Israel J. Math. 219 (1), 331–351, 2017.
  • [42] G. Molica Bisci, D. Repovs and R. Servadei, Nonlinear problems on the Sierpiński gasket, J. Math. Anal. Appl. 452 (2), 883–895, 2017.
  • [43] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMSReg. Conf. Ser. Math. 65, 1986.
  • [44] R. Rammal, A spectrum of harmonic excitations on fractals, J. Phy. Lett. 45, 191– 206, 1984.
  • [45] D. Stancu-Dumitru, Two nontrivial weak solutions for the Dirichlet problem on the Sierpiński gasket, Bull. Aust. Math. Soc. 85, 395–414, 2012.
  • [46] R.S. Strichartz, Analysis on fractals, Notices Amer. Math. Soc. 46, 1199–1208, 1999.
  • [47] R.S. Strichartz, Differential Equations on Fractals, A. Tutorial (ed.), Princeton Uni- versity Press, Princeton, 2006.
  • [48] A. Teplyaev, Spectral Analysis on Infinite Sierpiński Gaskets J. Func. Anal. 159, 537–567, 1998.
  • [49] G.Q. Zhang, X.P. Liu and S.Y. Liu, Remarks on a class of quasilinear elliptic systems involving the $(p,q)$-Laplacian, Electron. J. Differ. Equ. 2005, 1–10, 2005.

Year 2019, Volume: 48 Issue: 5, 1461 - 1478, 08.10.2019

Haiffa Muhsan B. Alrikabi Ghasem A. Afrouzi , Mohsen Alimohammady

Abstract

References

  • [1] S. Alexander, Some properties of the spectrum of the Sierpiński gasket in a magnetic field, Phys. Rev. B, 29, 5504–5508, 1984.
  • [2] M.T. Barlow and J. Kigami, Localized eigenfunctions of the Laplacian on p.c.f. self- similar sets, J. London Math. Soc. 56 (2), 320–332, 1997.
  • [3] T. Bartsch and D.G. de Figueiredo, Infinitely many solutions of nonlinear elliptic systems, Progr. Nonlinear Diff. Equ. Appl. 35, 51–67, 1999.
  • [4] A. Bensedik and M. Bouchekif, On certain nonlinear elliptic systems with indefinite terms, Electron. J. Differ. Equ. 83, 1–16, 2002.
  • [5] L. Boccardo and D.G. de Figueiredo, Some remarks on a system of quasilinear elliptic equations, Nonlinear Differ. Equ. Appl. 9, 309–323, 2002.
  • [6] B. Bockelman and R.S. Strichartz, Partial differential equations on products of Sierpiński gaskets, Indiana Univ. Math. J. 56, 1361–1375, 2007.
  • [7] M. Bohner, S. Heidarkhani, A. Salari and G. Caristi, Existence of three solutions for impulsive multi-point boundary value problems, Opuscula Math. 37 (3), 353–379, 2017.
  • [8] G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal. TMA, 75, 2992–3007, 2012.
  • [9] G. Bonanno, Relations between the mountain pass theorem and local minima, Adv. Nonlinear Anal. 1, 205–220, 2012.
  • [10] G. Bonanno and A. Chinnì, Existence and multiplicity of weak solutions for elliptic Dirichlet problems with variable exponent, J. Math. Anal. Appl. 418, 812–827, 2014.
  • [11] G. Bonanno and S.A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal. 89, 1–10, 2010.
  • [12] G. Bonanno, G. Molica Bisci and D. O’Regan, Infinitely many weak solutions for a class of quasilinear elliptic systems, Math. Comput. Modelling, 52, 152–160, 2010.
  • [13] G. Bonanno, G. Molica Bisci and V.D. Rădulescu, Infinitely many solutions for a class of nonlinear elliptic problems on fractals, C. R. Math. Acad. Sci. Paris, 350 (3-4), 187–191, 2012.
  • [14] G. Bonanno, G. Molica Bisci and V.D. Rădulescu, Variational analysis for a nonlinear elliptic problem on the Sierpiński Gasket, ESAIM Control Optim. Calc. Var. 18 (4), 941–953, 2012.
  • [15] G. Bonanno, G. Molica Bisci and V.D. Rădulescu, Qualitative Analysis of Gradient- Type Systems with Oscillatory Nonlinearities on the Sierpiński Gasket, Chin. Ann. Math. 34 (3), 381–398, 2013.
  • [16] Y. Bozhkova and E. Mitidieri, Existence of multiple solutions for quasilinear systems via fibering method, J. Differential Equations, 190, 239–267, 2003.
  • [17] B.E. Breckner, V.D. Rădulescu and C. Varga, Infinitely many solutions for the Dirich- let problem on the Sierpiński gasket, Anal. Appl. 9 (3), 235–248, 2011.
  • [18] B.E. Breckner, D. Repovš and C. Varga, On the existence of three solutions for the Dirichlet problem on the Sierpiński gasket, Nonlinear Anal. 73, 2980–2990, 2010.
  • [19] Ph. Clément, D.G. de Figueiredo and E. Mitidieri, Positive solutions of semilinear elliptic systems, Comm. Partial Differential Equations, 17, 923–940, 1992.
  • [20] D.G. de Figueiredo, Semilinear elliptic systems: a survey of superlinear problems, Resenhas IME-USP, 2, 373–391, 1996.
  • [21] K.J. Falconer, Semilinear PDEs on self-similar fractals, Commun. Math. Phys. 206, 235–245, 1999.
  • [22] K.J. Falconer and J. Hu, Nonlinear elliptical equations on the Sierpiński gasket, J. Math. Anal. Appl. 240, 552–573, 1999.
  • [23] M. Ferrara, G. Molica Bisci and D. Repovs, Existence results for nonlinear elliptic problems on fractal domains, Adv. Nonlinear Anal. 5 (1), 75–84, 2016.
  • [24] M. Fukushima and T. Shima, On a spectral analysis for the Sierpiński gasket, Poten- tial Anal. 1, 1–35, 1992.
  • [25] M. Ghergu and V. Rˇadulescu, Nonlinear PDEs. Mathematical models in biology, chemistry and population genetics, Springer Monographs in Mathematics, Springer, Heidelberg, 2012.
  • [26] S. Goldstein, Random walks and diffusions on fractals, Percolation Theory and Er- godic Theory of Infinite Particle Systems, H. Kesten (ed.), 121–129, IMA Math. Appl., 8, Springer-Verlag, New York, 1987.
  • [27] J.R. Graef, S. Heidarkhani and L. Kong, Multiple solutions for systems of multi-point boundary value problems, Opuscula Math. 33 (2), 293–306, 2013.
  • [28] J.R. Graef, S. Heidarkhani and L. Kong, Infinitely Many Solutions for Systems of Sturm-Liouville Boundary Value Problems, Results. Math. 66, 327–341, 2014.
  • [29] J.R. Graef, S. Heidarkhani and L. Kong, Nontrivial solutions for systems of Sturm- Liouville boundary value problems, Differ. Equ. Appl. 6 (2), 255–265, 2014.
  • [30] J.R. Graef, S. Heidarkhani and L. Kong, Multiple solutions for systems of Sturm- Liouville boundary value problems, Mediterr. J. Math. 13, 1625–1640, 2016.
  • [31] J.R. Graef, S. Heidarkhani and L. Kong, Multiple periodic solutions for perturbed second-order impulsive Hamiltonian systems, Int. J. Pure Appl. Math. 109 (1), 85– 104, 2016.
  • [32] D.D. Hai and R. Shivaji, An existence result on positive solutions for a class of p- Laplacian systems, Nonlinear Anal., 56, 1007–1010, 2004.
  • [33] S. Heidarkhani, Non-trivial solutions for two-point boundary-value problems of fourth- order Sturm-Liouville type equations, Electron. J. Diff. Equ. 2012 (27), 1–9, 2012.
  • [34] S. Heidarkhani, A. Salari and G. Caristi, Kirchhoff-type second-order impulsive dif- ferential equations on the half-line, preprint.
  • [35] J. Hu, Multiple solutions for a class of nonlinear elliptic equations on the Sierpiński gasket, Sci. China Ser. A, 47, 772–786, 2004.
  • [36] C. Hua and H. Zhenya, Semilinear elliptic equations on fractal sets, Acta Math. Sci. Ser. B, 29 (2), 232–242, 2009.
  • [37] J.E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30, 713–747, 1981.
  • [38] S. Kusuoka, A diffusion process on a fractal, Probabilistic Methods in Mathematical Physics, Katata/Kyoto, 1985, 251-274, Academic Press, Boston, 1987.
  • [39] S.M. Kozlov, Harmonization and homogenization on fractals, Communications Math. Phys. 153, 339–357, 1993.
  • [40] G. Molica Bisci and V. Rˇadulescu, A characterization for elliptic problems on fractal sets, Proc. Amer. Math. Soc. 143 (7), 2959–2968, 2015.
  • [41] G. Molica Bisci and V. Rˇadulescu, A sharp eigenvalue theorem for fractional elliptic equations, Israel J. Math. 219 (1), 331–351, 2017.
  • [42] G. Molica Bisci, D. Repovs and R. Servadei, Nonlinear problems on the Sierpiński gasket, J. Math. Anal. Appl. 452 (2), 883–895, 2017.
  • [43] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMSReg. Conf. Ser. Math. 65, 1986.
  • [44] R. Rammal, A spectrum of harmonic excitations on fractals, J. Phy. Lett. 45, 191– 206, 1984.
  • [45] D. Stancu-Dumitru, Two nontrivial weak solutions for the Dirichlet problem on the Sierpiński gasket, Bull. Aust. Math. Soc. 85, 395–414, 2012.
  • [46] R.S. Strichartz, Analysis on fractals, Notices Amer. Math. Soc. 46, 1199–1208, 1999.
  • [47] R.S. Strichartz, Differential Equations on Fractals, A. Tutorial (ed.), Princeton Uni- versity Press, Princeton, 2006.
  • [48] A. Teplyaev, Spectral Analysis on Infinite Sierpiński Gaskets J. Func. Anal. 159, 537–567, 1998.
  • [49] G.Q. Zhang, X.P. Liu and S.Y. Liu, Remarks on a class of quasilinear elliptic systems involving the $(p,q)$-Laplacian, Electron. J. Differ. Equ. 2005, 1–10, 2005.
There are 49 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Haiffa Muhsan B. Alrikabi This is me 0000-0002-9615-8480

Ghasem A. Afrouzi 0000-0001-8794-3594

Mohsen Alimohammady This is me 0000-0001-8358-9962

Publication Date October 8, 2019
Published in Issue Year 2019 Volume: 48 Issue: 5

Cite

APA Alrikabi, H. M. B., Afrouzi, G. A., & Alimohammady, M. (2019). Existence and multiplicity of weak solutions for gradient-type systems with oscillatory nonlinearities on the Sierpiński gasket. Hacettepe Journal of Mathematics and Statistics, 48(5), 1461-1478.
AMA Alrikabi HMB, Afrouzi GA, Alimohammady M. Existence and multiplicity of weak solutions for gradient-type systems with oscillatory nonlinearities on the Sierpiński gasket. Hacettepe Journal of Mathematics and Statistics. October 2019;48(5):1461-1478.
Chicago Alrikabi, Haiffa Muhsan B., Ghasem A. Afrouzi, and Mohsen Alimohammady. “Existence and Multiplicity of Weak Solutions for Gradient-Type Systems With Oscillatory Nonlinearities on the Sierpiński Gasket”. Hacettepe Journal of Mathematics and Statistics 48, no. 5 (October 2019): 1461-78.
EndNote Alrikabi HMB, Afrouzi GA, Alimohammady M (October 1, 2019) Existence and multiplicity of weak solutions for gradient-type systems with oscillatory nonlinearities on the Sierpiński gasket. Hacettepe Journal of Mathematics and Statistics 48 5 1461–1478.
IEEE H. M. B. Alrikabi, G. A. Afrouzi, and M. Alimohammady, “Existence and multiplicity of weak solutions for gradient-type systems with oscillatory nonlinearities on the Sierpiński gasket”, Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 5, pp. 1461–1478, 2019.
ISNAD Alrikabi, Haiffa Muhsan B. et al. “Existence and Multiplicity of Weak Solutions for Gradient-Type Systems With Oscillatory Nonlinearities on the Sierpiński Gasket”. Hacettepe Journal of Mathematics and Statistics 48/5 (October 2019), 1461-1478.
JAMA Alrikabi HMB, Afrouzi GA, Alimohammady M. Existence and multiplicity of weak solutions for gradient-type systems with oscillatory nonlinearities on the Sierpiński gasket. Hacettepe Journal of Mathematics and Statistics. 2019;48:1461–1478.
MLA Alrikabi, Haiffa Muhsan B. et al. “Existence and Multiplicity of Weak Solutions for Gradient-Type Systems With Oscillatory Nonlinearities on the Sierpiński Gasket”. Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 5, 2019, pp. 1461-78.
Vancouver Alrikabi HMB, Afrouzi GA, Alimohammady M. Existence and multiplicity of weak solutions for gradient-type systems with oscillatory nonlinearities on the Sierpiński gasket. Hacettepe Journal of Mathematics and Statistics. 2019;48(5):1461-78.

https://dergipark.org.tr/en/pub/hujms