• Previous Article
    Bifurcation of limit cycles from a Li$ \acute{E} $nard system with asymmetric figure eight-loop case
  • DCDS-S Home
  • This Issue
  • Next Article
    Asymptotic spreading for Fisher-KPP reaction-diffusion equations with heterogeneous shifting diffusivity
doi: 10.3934/dcdss.2022105
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Weak solvability of nonlinear elliptic equations involving variable exponents

1. 

Laboratory LAMA, Sidi Mohamed Ben Abdellah University, National School of Applied Sciences, Fez, Morocco

2. 

Laboratory LAMA, Department of Mathematics, Sidi Mohamed Ben Abdellah University, Faculty of Sciences Dhar El Mahraz, B.P 1796 Atlas Fez, Morocco

3. 

Dipartimento di Matematica e Informatica, Universitá di Catania, Catania, Italy

4. 

RUDN University, 6 Miklukho-Maklay St, 117198, Moscow, Russia

* Corresponding author: Maria Alessandra Ragusa

Received  February 2022 Early access April 2022

Fund Project: The fourth author is supported by RUDN University Strategic Academic Leadership Program and P.R.I.N. 2019

We are concerned with the study of the existence and multiplicity of solutions for Dirichlet boundary value problems, involving the $ ( p( m ), \, q( m ) )- $ equation and the nonlinearity is superlinear but does not fulfil the Ambrossetti-Rabinowitz condition in the framework of Sobolev spaces with variable exponents in a complete manifold. The main results are proved using the mountain pass theorem and Fountain theorem with Cerami sequences. Moreover, an example of a $ ( p( m ), \, q( m ) ) $ equation that highlights the applicability of our theoretical results is also provided.

Citation: Ahmed Aberqi, Jaouad Bennouna, Omar Benslimane, Maria Alessandra Ragusa. Weak solvability of nonlinear elliptic equations involving variable exponents. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022105
References:
[1]

A. Aberqi, J. Bennouna, O. Benslimane and M. A. Ragusa, Existence Results for double phase problem in Sobolev-Orlicz spaces with variable exponents in Complete Manifold, Mediterranean Journal of Mathematics, (2022).

[2]

A. AberqiJ. BennounaM. Elmassoudi and M. Hammoumi, Existence and uniqueness of a renormalized solution of parabolic problems in Orlicz spaces, Monatsh. Math., 189 (2019), 195-219.  doi: 10.1007/s00605-018-01260-8.

[3]

A. AberqiJ. BennounaM. Mekkour and H. Redwane, Nonlinear parabolic inequalities with lower order terms, Appl. Anal., 96 (2017), 2102-2117.  doi: 10.1080/00036811.2016.1205186.

[4]

A. Aberqi, O. Benslimane, A. Ouaziz and D. D. Repovš, On a new fractional Sobolev space with variable exponent on complete manifolds, Bound Value Probl, 2022 (2022), Paper No. 7, 20 pp. doi: 10.1186/s13661-022-01590-5.

[5]

T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampére Equations, Grundlehren der mathematischen Wissenschaften, 252. Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5734-9.

[6]

E. AzroulA. Benkirane and M. Shimi, On a nonlocal problem involving the fractional $ p (x, .) $-Laplacian satisfying Cerami condition, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 3479-2495.  doi: 10.3934/dcdss.2020425.

[7]

P. BaroniM. Colombo and G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal., 121 (2015), 206-222.  doi: 10.1016/j.na.2014.11.001.

[8]

O. BenslimaneA. Aberqi and J. Bennouna, Existence and Uniqueness of Weak solution of $p(x)$-laplacian in Sobolev spaces with variable exponents in complete manifolds, Filomat, 35 (2021), 1453-1463.  doi: 10.2298/FIL2105453B.

[9]

O. BenslimaneA. Aberqi and J. Bennouna, The existence and uniqueness of an entropy solution to unilateral Orlicz anisotropic equations in an unbounded domain, Axioms, 9 (2020), 109. 

[10]

O. BenslimaneA. Aberqi and J. Bennouna, Existence and uniqueness of entropy solution of a nonlinear elliptic equation in anisotropic Sobolev–Orlicz space, Rend. Circ. Mat. Palermo, 70 (2021), 1579-1608.  doi: 10.1007/s12215-020-00577-4.

[11]

O. Benslimane, A. Aberqi and J. Bennouna, On some nonlinear anisotropic elliptic equations in anisotropic Orlicz space, Arab Journal of Mathematical Sciences, (2021). doi: 10.1108/AJMS-12-2020-0133.

[12]

O. BenslimaneA. Aberqi and J. Bennouna, Existence results for double phase obstacle problems with variable exponents, Journal of Elliptic and Parabolic Equations, 7 (2021), 875-890.  doi: 10.1007/s41808-021-00122-z.

[13]

L. BoccardoT. Gallouët and J. L. Vazquez, Nonlinear elliptic equations in $ \mathbb{R}^{N} $ without growth restrictions on the data, J. Differential Equations, 105 (1993), 334-363.  doi: 10.1006/jdeq.1993.1092.

[14]

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

[15]

Y. ChenS. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM journal on Applied Mathematics, 66 (2006), 1383-1406.  doi: 10.1137/050624522.

[16]

M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal., 215 (2015), 443-496.  doi: 10.1007/s00205-014-0785-2.

[17]

M. GaczkowskiP. Górka and D. J. Pon, Sobolev spaces with variable exponents on complete manifolds, J. Funct. Anal., 270 (2016), 1379-1415.  doi: 10.1016/j.jfa.2015.09.008.

[18]

L. Guo, The Dirichlrt problems for nonlinear elliptic equations with variable exponents on Riemannian manifolds, J. Appl. Anal. Comput., 5 (2015), 562-569.  doi: 10.11948/2015043.

[19]

A. K. Gushchin, The Dirichlet problem for a second-order elliptic equation with an $L^{p}$ boundary function, Sb. Math., 203 (2012), 1-27.  doi: 10.1134/S1064562411020281.

[20]

P. Gwiazda and A. Świerczewska-Gwiazda, On non-Newtonian fluids with a property of rapid thickening under different stimulus, Math. Models Methods Appl. Sci., 18 (2008), 1073-1092.  doi: 10.1142/S0218202508002954.

[21]

E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lecture Notes in Mathematics, 5. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 1999.

[22]

E. J. HurtadoO. H. Miyagaki and R. S. Rodrigues, Existence and multiplicity of solutions for a class of elliptic equations without Ambrosetti-Rabinowitz type conditions, J. Dynam. Differential Equations, 30 (2018), 405-432.  doi: 10.1007/s10884-016-9542-6.

[23]

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

[24]

G. I. Laptev, Existence of solutions of certain quasilinear elliptic equations in $ \mathbb{R}^{ N } $ without conditions at infinity, J. Math. Sci. (N.Y.), 150 (2008), 2384-2394.  doi: 10.1007/s10958-008-0137-6.

[25]

S. B. Liu and S. J. Li, Infinitely many solutions for a superlinear elliptic equation, Acta Math. Sinica, 46 (2003), 625-630. 

[26]

M. A. Ragusa and A. Tachikawa, Regularity for minimizers for functionals of double phase with variable exponents, Adv. Nonlinear Anal., 9 (2020), 710-728.  doi: 10.1515/anona-2020-0022.

[27]

M. Ruzicka, Modeling, mathematical and numerical analysis of electrorheological fluids, Appl. Math., 49 (2004), 565-609.  doi: 10.1007/s10492-004-6432-8.

[28]

N. S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1968), 265-274. 

[29]

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.

[30]

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.

[31]

V. V. Zhikov, On density of smooth functions in Sobolev–Orlich spaces, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 310 (2004), 67-81.  doi: 10.1007/s10958-005-0497-0.

[32]

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

show all references

References:
[1]

A. Aberqi, J. Bennouna, O. Benslimane and M. A. Ragusa, Existence Results for double phase problem in Sobolev-Orlicz spaces with variable exponents in Complete Manifold, Mediterranean Journal of Mathematics, (2022).

[2]

A. AberqiJ. BennounaM. Elmassoudi and M. Hammoumi, Existence and uniqueness of a renormalized solution of parabolic problems in Orlicz spaces, Monatsh. Math., 189 (2019), 195-219.  doi: 10.1007/s00605-018-01260-8.

[3]

A. AberqiJ. BennounaM. Mekkour and H. Redwane, Nonlinear parabolic inequalities with lower order terms, Appl. Anal., 96 (2017), 2102-2117.  doi: 10.1080/00036811.2016.1205186.

[4]

A. Aberqi, O. Benslimane, A. Ouaziz and D. D. Repovš, On a new fractional Sobolev space with variable exponent on complete manifolds, Bound Value Probl, 2022 (2022), Paper No. 7, 20 pp. doi: 10.1186/s13661-022-01590-5.

[5]

T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampére Equations, Grundlehren der mathematischen Wissenschaften, 252. Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5734-9.

[6]

E. AzroulA. Benkirane and M. Shimi, On a nonlocal problem involving the fractional $ p (x, .) $-Laplacian satisfying Cerami condition, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 3479-2495.  doi: 10.3934/dcdss.2020425.

[7]

P. BaroniM. Colombo and G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal., 121 (2015), 206-222.  doi: 10.1016/j.na.2014.11.001.

[8]

O. BenslimaneA. Aberqi and J. Bennouna, Existence and Uniqueness of Weak solution of $p(x)$-laplacian in Sobolev spaces with variable exponents in complete manifolds, Filomat, 35 (2021), 1453-1463.  doi: 10.2298/FIL2105453B.

[9]

O. BenslimaneA. Aberqi and J. Bennouna, The existence and uniqueness of an entropy solution to unilateral Orlicz anisotropic equations in an unbounded domain, Axioms, 9 (2020), 109. 

[10]

O. BenslimaneA. Aberqi and J. Bennouna, Existence and uniqueness of entropy solution of a nonlinear elliptic equation in anisotropic Sobolev–Orlicz space, Rend. Circ. Mat. Palermo, 70 (2021), 1579-1608.  doi: 10.1007/s12215-020-00577-4.

[11]

O. Benslimane, A. Aberqi and J. Bennouna, On some nonlinear anisotropic elliptic equations in anisotropic Orlicz space, Arab Journal of Mathematical Sciences, (2021). doi: 10.1108/AJMS-12-2020-0133.

[12]

O. BenslimaneA. Aberqi and J. Bennouna, Existence results for double phase obstacle problems with variable exponents, Journal of Elliptic and Parabolic Equations, 7 (2021), 875-890.  doi: 10.1007/s41808-021-00122-z.

[13]

L. BoccardoT. Gallouët and J. L. Vazquez, Nonlinear elliptic equations in $ \mathbb{R}^{N} $ without growth restrictions on the data, J. Differential Equations, 105 (1993), 334-363.  doi: 10.1006/jdeq.1993.1092.

[14]

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

[15]

Y. ChenS. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM journal on Applied Mathematics, 66 (2006), 1383-1406.  doi: 10.1137/050624522.

[16]

M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal., 215 (2015), 443-496.  doi: 10.1007/s00205-014-0785-2.

[17]

M. GaczkowskiP. Górka and D. J. Pon, Sobolev spaces with variable exponents on complete manifolds, J. Funct. Anal., 270 (2016), 1379-1415.  doi: 10.1016/j.jfa.2015.09.008.

[18]

L. Guo, The Dirichlrt problems for nonlinear elliptic equations with variable exponents on Riemannian manifolds, J. Appl. Anal. Comput., 5 (2015), 562-569.  doi: 10.11948/2015043.

[19]

A. K. Gushchin, The Dirichlet problem for a second-order elliptic equation with an $L^{p}$ boundary function, Sb. Math., 203 (2012), 1-27.  doi: 10.1134/S1064562411020281.

[20]

P. Gwiazda and A. Świerczewska-Gwiazda, On non-Newtonian fluids with a property of rapid thickening under different stimulus, Math. Models Methods Appl. Sci., 18 (2008), 1073-1092.  doi: 10.1142/S0218202508002954.

[21]

E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lecture Notes in Mathematics, 5. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 1999.

[22]

E. J. HurtadoO. H. Miyagaki and R. S. Rodrigues, Existence and multiplicity of solutions for a class of elliptic equations without Ambrosetti-Rabinowitz type conditions, J. Dynam. Differential Equations, 30 (2018), 405-432.  doi: 10.1007/s10884-016-9542-6.

[23]

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

[24]

G. I. Laptev, Existence of solutions of certain quasilinear elliptic equations in $ \mathbb{R}^{ N } $ without conditions at infinity, J. Math. Sci. (N.Y.), 150 (2008), 2384-2394.  doi: 10.1007/s10958-008-0137-6.

[25]

S. B. Liu and S. J. Li, Infinitely many solutions for a superlinear elliptic equation, Acta Math. Sinica, 46 (2003), 625-630. 

[26]

M. A. Ragusa and A. Tachikawa, Regularity for minimizers for functionals of double phase with variable exponents, Adv. Nonlinear Anal., 9 (2020), 710-728.  doi: 10.1515/anona-2020-0022.

[27]

M. Ruzicka, Modeling, mathematical and numerical analysis of electrorheological fluids, Appl. Math., 49 (2004), 565-609.  doi: 10.1007/s10492-004-6432-8.

[28]

N. S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1968), 265-274. 

[29]

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.

[30]

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.

[31]

V. V. Zhikov, On density of smooth functions in Sobolev–Orlich spaces, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 310 (2004), 67-81.  doi: 10.1007/s10958-005-0497-0.

[32]

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

[1]

Genni Fragnelli, Paolo Nistri, Duccio Papini. Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 35-64. doi: 10.3934/dcds.2011.31.35

[2]

Rui Huang, Yifu Wang, Yuanyuan Ke. Existence of non-trivial nonnegative periodic solutions for a class of degenerate parabolic equations with nonlocal terms. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 1005-1014. doi: 10.3934/dcdsb.2005.5.1005

[3]

Wouter Castryck, Marco Streng, Damiano Testa. Curves in characteristic $2$ with non-trivial $2$-torsion. Advances in Mathematics of Communications, 2014, 8 (4) : 479-495. doi: 10.3934/amc.2014.8.479

[4]

Fernando Alcalde Cuesta, Françoise Dal'Bo, Matilde Martínez, Alberto Verjovsky. Minimality of the horocycle flow on laminations by hyperbolic surfaces with non-trivial topology. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4619-4635. doi: 10.3934/dcds.2016001

[5]

Kung-Ching Chang, Zhi-Qiang Wang, Tan Zhang. On a new index theory and non semi-trivial solutions for elliptic systems. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 809-826. doi: 10.3934/dcds.2010.28.809

[6]

Inbo Sim, Yun-Ho Kim. Existence of solutions and positivity of the infimum eigenvalue for degenerate elliptic equations with variable exponents. Conference Publications, 2013, 2013 (special) : 695-707. doi: 10.3934/proc.2013.2013.695

[7]

Fernando Alcalde Cuesta, Françoise Dal'Bo, Matilde Martínez, Alberto Verjovsky. Corrigendum to "Minimality of the horocycle flow on laminations by hyperbolic surfaces with non-trivial topology". Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4585-4586. doi: 10.3934/dcds.2017196

[8]

Vyacheslav Grines, Dmitrii Mints. On decomposition of ambient surfaces admitting $ A $-diffeomorphisms with non-trivial attractors and repellers. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3557-3568. doi: 10.3934/dcds.2022024

[9]

Yanfang Peng, Jing Yang. Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents. Communications on Pure and Applied Analysis, 2015, 14 (2) : 439-455. doi: 10.3934/cpaa.2015.14.439

[10]

Anna Maria Candela, J.L. Flores, M. Sánchez. A quadratic Bolza-type problem in a non-complete Riemannian manifold. Conference Publications, 2003, 2003 (Special) : 173-181. doi: 10.3934/proc.2003.2003.173

[11]

Vy Khoi Le. On the existence of nontrivial solutions of inequalities in Orlicz-Sobolev spaces. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 809-818. doi: 10.3934/dcdss.2012.5.809

[12]

Shengbing Deng, Zied Khemiri, Fethi Mahmoudi. On spike solutions for a singularly perturbed problem in a compact riemannian manifold. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2063-2084. doi: 10.3934/cpaa.2018098

[13]

Minbo Yang, Fukun Zhao, Shunneng Zhao. Classification of solutions to a nonlocal equation with doubly Hardy-Littlewood-Sobolev critical exponents. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5209-5241. doi: 10.3934/dcds.2021074

[14]

Giovany M. Figueiredo, Tarcyana S. Figueiredo-Sousa, Cristian Morales-Rodrigo, Antonio Suárez. Existence of positive solutions of an elliptic equation with local and nonlocal variable diffusion coefficient. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3689-3711. doi: 10.3934/dcdsb.2018311

[15]

E. Camouzis, H. Kollias, I. Leventides. Stable manifold market sequences. Journal of Dynamics and Games, 2018, 5 (2) : 165-185. doi: 10.3934/jdg.2018010

[16]

Marco Ghimenti, A. M. Micheletti. Non degeneracy for solutions of singularly perturbed nonlinear elliptic problems on symmetric Riemannian manifolds. Communications on Pure and Applied Analysis, 2013, 12 (2) : 679-693. doi: 10.3934/cpaa.2013.12.679

[17]

Mostafa Bendahmane, Kenneth Hvistendahl Karlsen, Mazen Saad. Nonlinear anisotropic elliptic and parabolic equations with variable exponents and $L^1$ data. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1201-1220. doi: 10.3934/cpaa.2013.12.1201

[18]

Peter E. Kloeden, Jacson Simsen. Pullback attractors for non-autonomous evolution equations with spatially variable exponents. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2543-2557. doi: 10.3934/cpaa.2014.13.2543

[19]

Qingfang Wang. The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2261-2281. doi: 10.3934/cpaa.2018108

[20]

Sabri Bahrouni, Hichem Ounaies. Embedding theorems in the fractional Orlicz-Sobolev space and applications to non-local problems. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2917-2944. doi: 10.3934/dcds.2020155

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (98)
  • HTML views (36)
  • Cited by (0)

[Back to Top]