• Previous Article
    Finite- and multi-dimensional state representations and some fundamental asymptotic properties of a family of nonlinear multi-population models for HIV/AIDS with ART treatment and distributed delays
  • DCDS-S Home
  • This Issue
  • Next Article
    On properties of similarity boundary of attractors in product dynamical systems
doi: 10.3934/dcdss.2020425

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

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  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.

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, 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. AzroulA. 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. BartoloV. 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 NezzaG. 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. KaufmannJ. 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. AzroulA. 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. BartoloV. 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 NezzaG. 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. KaufmannJ. 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

[1]

Said Taarabti. Positive solutions for the $ p(x)- $Laplacian : Application of the Nehari method. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021029

[2]

Alfonso Castro, Jorge Cossio, Sigifredo Herrón, Carlos Vélez. Infinitely many radial solutions for a $ p $-Laplacian problem with indefinite weight. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021058

[3]

Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3211-3240. doi: 10.3934/dcds.2020403

[4]

Anhui Gu. Weak pullback mean random attractors for non-autonomous $ p $-Laplacian equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3863-3878. doi: 10.3934/dcdsb.2020266

[5]

Jong Yoon Hyun, Yoonjin Lee, Yansheng Wu. Connection of $ p $-ary $ t $-weight linear codes to Ramanujan Cayley graphs with $ t+1 $ eigenvalues. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020133

[6]

Khalid Latrach, Hssaine Oummi, Ahmed Zeghal. Existence results for nonlinear mono-energetic singular transport equations: $ L^p $-spaces. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021028

[7]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2635-3652. doi: 10.3934/dcds.2020378

[8]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113

[9]

Yishui Wang, Dongmei Zhang, Peng Zhang, Yong Zhang. Local search algorithm for the squared metric $ k $-facility location problem with linear penalties. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2013-2030. doi: 10.3934/jimo.2020056

[10]

Chenchen Wu, Wei Lv, Yujie Wang, Dachuan Xu. Approximation Algorithm for Spherical $ k $-Means Problem with Penalty. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021067

[11]

Lin Yang, Yejuan Wang, Tomás Caraballo. Regularity of global attractors and exponential attractors for $ 2 $D quasi-geostrophic equations with fractional dissipation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021093

[12]

Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002

[13]

Jennifer D. Key, Bernardo G. Rodrigues. Binary codes from $ m $-ary $ n $-cubes $ Q^m_n $. Advances in Mathematics of Communications, 2021, 15 (3) : 507-524. doi: 10.3934/amc.2020079

[14]

Chaoqian Li, Yajun Liu, Yaotang Li. Note on $ Z $-eigenvalue inclusion theorems for tensors. Journal of Industrial & Management Optimization, 2021, 17 (2) : 687-693. doi: 10.3934/jimo.2019129

[15]

Xiaoni Chi, Zhongping Wan, Zijun Hao. A full-modified-Newton step $ O(n) $ infeasible interior-point method for the special weighted linear complementarity problem. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021082

[16]

Dean Crnković, Nina Mostarac, Bernardo G. Rodrigues, Leo Storme. $ s $-PD-sets for codes from projective planes $ \mathrm{PG}(2,2^h) $, $ 5 \leq h\leq 9 $. Advances in Mathematics of Communications, 2021, 15 (3) : 423-440. doi: 10.3934/amc.2020075

[17]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195

[18]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447

[19]

Raj Kumar, Maheshanand Bhaintwal. Duadic codes over $ \mathbb{Z}_4+u\mathbb{Z}_4 $. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020135

[20]

Brian Ryals, Robert J. Sacker. Bifurcation in the almost periodic $ 2 $D Ricker map. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021089

2019 Impact Factor: 1.233

Article outline

[Back to Top]