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October  2021, 14(10): 3479-3495. doi: 10.3934/dcdss.2020425

On a nonlocal problem involving the fractional $ p(x,.) $-Laplacian satisfying Cerami condition

Elhoussine Azroul , Abdelmoujib Benkirane and  and Mohammed Shimi ,

Sidi Mohamed Ben Abdellah University, Faculty of Sciences Dhar El Mahraz, Laboratory of Mathematical Analysis and Applications, Fez, Morocco, B.P. 1796 Fez-Atlas, 30003 MOROCCO

* Corresponding author: Mohammed Shimi

Received  March 2020 Revised  June 2020 Published  October 2021 Early access  September 2020

The present paper deals with the existence and multiplicity of solutions for a class of fractional $ p(x,.) $-Laplacian problems with the nonlocal Dirichlet boundary data, where the nonlinearity is superlinear but does not satisfy the usual Ambrosetti-Rabinowitz condition. To overcome the difficulty that the Palais-Smale sequences of the Euler-Lagrange functional may be unbounded, we consider the Cerami sequences. The main results are established by means of mountain pass theorem and Fountain theorem with Cerami condition.

Keywords: Fractional $ p(x,.) $-Laplacian operator, Nonlocal problem, Cerami condition, Mountain pass theorem, Fountain theorem.
Mathematics Subject Classification: Primary:35R11, 47G30;Secondary:35S15, 35A15.
Citation: Elhoussine Azroul, Abdelmoujib Benkirane, and Mohammed Shimi. On a nonlocal problem involving the fractional $ p(x,.) $-Laplacian satisfying Cerami condition. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3479-3495. doi: 10.3934/dcdss.2020425
References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-341.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

E. Azroul, A. Benkirane, M. Shimi and M. Srati, On a class of fractional $p(x)$-Kirchhoff type problems, Applicable Analysis, 2019 (2019). doi: 10.1080/00036811.2019.1603372.  Google Scholar

[3]

E. Azroul, A. Benkirane and M. Shimi, Eigenvalue problems involving the fractional $p(x)$-Laplacian operator, Adv. Oper. Theory, 4 (2019), 539-555.  doi: 10.15352/aot.1809-1420.  Google Scholar

[4]

E. Azroul, A. Benkirane, A. Boumazourh and M. Shimi, Existence results for fractional $p(x, .)$-Laplacian problem via the Nehari manifold approach, Applied Mathematics and Optimization, (2020). doi: 10.1007/s00245-020-09686-z.  Google Scholar

[5]

E. Azroul, A. Benkirane, A. Boumazourh and M. Srati, Three solutions for a nonlocal fractional $p$-Kirchhoff type elliptic system, Applicable Analysis, (2019). doi: 10.1080/00036811.2019.1670347.  Google Scholar

[6]

A. Bahrouni and V. Rǎdulescu, On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent, Discrete Contin. Dyn. Syst. S, 11 (2018), 379-389.  doi: 10.3934/dcdss.2018021.  Google Scholar

[7]

P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with "strong" resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012.  doi: 10.1016/0362-546X(83)90115-3.  Google Scholar

[8]

G. Cerami, An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend., 112 (1978), 332-336.   Google Scholar

[9]

N. T. Chung and H. Q. Toan, On a class of fractional Laplacian problems with variable exponents and indefinite weights, Collectanea Mathematica, 2019 (2019), 1-15.  doi: 10.1007/s13348-019-00254-5.  Google Scholar

[10]

R. De La Llave and E. Valdinoci, Symmetry for a Dirichlet Neumann problem arising in water waves, Math. Res. Lett., 16 (2009), 909-918.  doi: 10.4310/MRL.2009.v16.n5.a13.  Google Scholar

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E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math, 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[12]

X. L. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl, 263 (2001), 424-446.  doi: 10.1006/jmaa.2000.7617.  Google Scholar

[13]

F. Fang and S. Liu, Nontrivial solutions of superlinear $p$-Laplacian equations, J. Math. Anal. Appl., 351 (2009), 138-146.  doi: 10.1016/j.jmaa.2008.09.064.  Google Scholar

[14]

M. Fǎrcǎşeanu, On an eigenvalue problem involving the fractional $(s, p)$-Laplacian, Fractional Calculus and Applied Analysis, 21 (2018), 94-103.  doi: 10.1515/fca-2018-0006.  Google Scholar

[15]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to Landesman-lazer type problem set on $\mathbb{R^N}$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809.  doi: 10.1017/S0308210500013147.  Google Scholar

[16]

U. Kaufmann, J. D. Rossi and R. Vidal, Fractional Sobolev spaces with variable exponents and fractional $p(x)$-Laplacians, Elec. J. Qual. Theor. Diff. Equa., 76 (2017), 1-10.  doi: 10.14232/ejqtde.2017.1.76.  Google Scholar

[17]

O. Kováčik and J. Rákosník, On Spaces $L^{p(x)}(\Omega)$ and $W^{m, p(x)}(\Omega)$, Czechoslovak Math. J., 41 (1991), 592–618, https://dml.cz/handle/10338.dmlcz/102493. Google Scholar

[18]

M. Kwasnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20, (2017), 7–51. doi: 10.1515/fca-2017-0002.  Google Scholar

[19]

N. Lam and G. Lu, Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti-Rabinowitz condition, J. Geom. Anal., 24 (2014), 118-143.  doi: 10.1007/s12220-012-9330-4.  Google Scholar

[20]

N Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett., 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[21]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.  doi: 10.1016/j.jfa.2009.01.020.  Google Scholar

[22]

L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and anomalous diffusion: A tutorial, in Order and Chaos (ed. T. Bountis), Patras University Press, 10, 2008, https://arxiv.org/pdf/0805.0419.pdf. Google Scholar

[23]

A. Zang, $p(x)$-Laplacian equations satisfying Cerami condition, J. Math. Anal. Appl., 337 (2008), 547-555.  doi: 10.1016/j.jmaa.2007.04.007.  Google Scholar

[24]

Q. Zhang and C. Zhao, Existence of strong solutions of a $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition, Comput. Math. Appl., 69 (2015), 1-12.  doi: 10.1016/j.camwa.2014.10.022.  Google Scholar

[25]

W. Zou, Variant fountain theorems and their applications, Manuscripta Math., 104 (2001), 343-358.  doi: 10.1007/s002290170032.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-341.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

E. Azroul, A. Benkirane, M. Shimi and M. Srati, On a class of fractional $p(x)$-Kirchhoff type problems, Applicable Analysis, 2019 (2019). doi: 10.1080/00036811.2019.1603372.  Google Scholar

[3]

E. Azroul, A. Benkirane and M. Shimi, Eigenvalue problems involving the fractional $p(x)$-Laplacian operator, Adv. Oper. Theory, 4 (2019), 539-555.  doi: 10.15352/aot.1809-1420.  Google Scholar

[4]

E. Azroul, A. Benkirane, A. Boumazourh and M. Shimi, Existence results for fractional $p(x, .)$-Laplacian problem via the Nehari manifold approach, Applied Mathematics and Optimization, (2020). doi: 10.1007/s00245-020-09686-z.  Google Scholar

[5]

E. Azroul, A. Benkirane, A. Boumazourh and M. Srati, Three solutions for a nonlocal fractional $p$-Kirchhoff type elliptic system, Applicable Analysis, (2019). doi: 10.1080/00036811.2019.1670347.  Google Scholar

[6]

A. Bahrouni and V. Rǎdulescu, On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent, Discrete Contin. Dyn. Syst. S, 11 (2018), 379-389.  doi: 10.3934/dcdss.2018021.  Google Scholar

[7]

P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with "strong" resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012.  doi: 10.1016/0362-546X(83)90115-3.  Google Scholar

[8]

G. Cerami, An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend., 112 (1978), 332-336.   Google Scholar

[9]

N. T. Chung and H. Q. Toan, On a class of fractional Laplacian problems with variable exponents and indefinite weights, Collectanea Mathematica, 2019 (2019), 1-15.  doi: 10.1007/s13348-019-00254-5.  Google Scholar

[10]

R. De La Llave and E. Valdinoci, Symmetry for a Dirichlet Neumann problem arising in water waves, Math. Res. Lett., 16 (2009), 909-918.  doi: 10.4310/MRL.2009.v16.n5.a13.  Google Scholar

[11]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math, 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[12]

X. L. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl, 263 (2001), 424-446.  doi: 10.1006/jmaa.2000.7617.  Google Scholar

[13]

F. Fang and S. Liu, Nontrivial solutions of superlinear $p$-Laplacian equations, J. Math. Anal. Appl., 351 (2009), 138-146.  doi: 10.1016/j.jmaa.2008.09.064.  Google Scholar

[14]

M. Fǎrcǎşeanu, On an eigenvalue problem involving the fractional $(s, p)$-Laplacian, Fractional Calculus and Applied Analysis, 21 (2018), 94-103.  doi: 10.1515/fca-2018-0006.  Google Scholar

[15]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to Landesman-lazer type problem set on $\mathbb{R^N}$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809.  doi: 10.1017/S0308210500013147.  Google Scholar

[16]

U. Kaufmann, J. D. Rossi and R. Vidal, Fractional Sobolev spaces with variable exponents and fractional $p(x)$-Laplacians, Elec. J. Qual. Theor. Diff. Equa., 76 (2017), 1-10.  doi: 10.14232/ejqtde.2017.1.76.  Google Scholar

[17]

O. Kováčik and J. Rákosník, On Spaces $L^{p(x)}(\Omega)$ and $W^{m, p(x)}(\Omega)$, Czechoslovak Math. J., 41 (1991), 592–618, https://dml.cz/handle/10338.dmlcz/102493. Google Scholar

[18]

M. Kwasnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20, (2017), 7–51. doi: 10.1515/fca-2017-0002.  Google Scholar

[19]

N. Lam and G. Lu, Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti-Rabinowitz condition, J. Geom. Anal., 24 (2014), 118-143.  doi: 10.1007/s12220-012-9330-4.  Google Scholar

[20]

N Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett., 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[21]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.  doi: 10.1016/j.jfa.2009.01.020.  Google Scholar

[22]

L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and anomalous diffusion: A tutorial, in Order and Chaos (ed. T. Bountis), Patras University Press, 10, 2008, https://arxiv.org/pdf/0805.0419.pdf. Google Scholar

[23]

A. Zang, $p(x)$-Laplacian equations satisfying Cerami condition, J. Math. Anal. Appl., 337 (2008), 547-555.  doi: 10.1016/j.jmaa.2007.04.007.  Google Scholar

[24]

Q. Zhang and C. Zhao, Existence of strong solutions of a $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition, Comput. Math. Appl., 69 (2015), 1-12.  doi: 10.1016/j.camwa.2014.10.022.  Google Scholar

[25]

W. Zou, Variant fountain theorems and their applications, Manuscripta Math., 104 (2001), 343-358.  doi: 10.1007/s002290170032.  Google Scholar

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