January  2022, 15(1): 213-227. doi: 10.3934/dcdss.2021015

Existence of solution for Kirchhoff type problem in Orlicz-Sobolev spaces Via Leray-Schauder's nonlinear alternative

1. 

Laboratory LIPIM, National School of Applied Sciences Khouribga, Sultan Moulay Slimane University, Morocco

2. 

Sidi Mohamed Ben Abdellah University, Faculty of Sciences Dhar El Mahraz, Laboratory of Mathematical Analysis and Applications, Fez, Morocco

* Corresponding author: yelhadfi@gmail.com

Received  September 2020 Revised  January 2021 Published  January 2022 Early access  January 2021

In this paper, we establish the existence of weak solution in Orlicz-Sobolev space for the following Kirchhoff type probelm
$ \begin{equation*} \left\{ \begin{array}{ll} -M\left( \int_{\Omega}\varPhi(|\nabla u|)dx\right) div(a(|\nabla u|)\nabla u) = f(x, u) \, in \, \, \, \, \Omega, \\ u = 0 \, \, \, \, on\, \, \, \, \, \, \, \, \, \, \partial \Omega, \end{array} \right. \end{equation*} $
where
$ \Omega $
is a bounded subset in
$ {\mathbb{R}}^N $
,
$ N\geq 1 $
with Lipschitz boundary
$ \partial \Omega. $
The used technical approach is mainly based on Leray-Shauder's non linear alternative.
Citation: Abdelaaziz Sbai, Youssef El Hadfi, Mohammed Srati, Noureddine Aboutabit. Existence of solution for Kirchhoff type problem in Orlicz-Sobolev spaces Via Leray-Schauder's nonlinear alternative. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 213-227. doi: 10.3934/dcdss.2021015
References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. 
[2]

G. A. AfrouziS. Heidarkhani and S. Shokooh, Infinitely many solutions for Steklov problems associated to non-homogeneous differential operators through Orlicz-Sobolev spaces, Complex Var. Elliptic Equ., 60 (2015), 1505-1521.  doi: 10.1080/17476933.2015.1031122.

[3]

C. O. AlvesF. S. J. A. Corrâa and T. F. Ma, Positive solutions for a quasilinear elliptic equations of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93.  doi: 10.1016/j.camwa.2005.01.008.

[4]

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.

[5]

E. AzroulA. Benkirane and M. Srati, Nonlocal eigenvalue type problem in fractional Orlicz-Sobolev space, Adv. Oper. Theory, 5 (2020), 1599-1617.  doi: 10.1007/s43036-020-00067-5.

[6]

E. Azroul, A. Benkirane and M. Shimi, Existence and multiplicity of solutions for fractional $p(x, .)-$Kirchhoff type problems in $\mathbb{R}^N, $, Applicable Analysis, (2019). doi: 10.1080/00036811.2019.1673373.

[7]

E. AzroulA. Benkirane and M. Srati, Existence of solutions for a nonlocal type problem in fractional Orlicz Sobolev spaces, Adv. Oper. Theory, 5 (2020), 1350-1375.  doi: 10.1007/s43036-020-00042-0.

[8]

A. Boumazourh and M. Srati, Leray-Schauder's solution for a nonlocal problem in a fractional Orlicz-Sobolev space, Moroccan J. of Pure and Appl. Anal. (MJPAA), (2020), 42–52.

[9]

F. Cammaroto and L. Vilasi, Multiple solutions for a Kirchhoff-type problem involving the $p(x)$-Laplacian operator, Nonlinear Anal., 74 (2011), 1841-1852.  doi: 10.1016/j.na.2010.10.057.

[10]

M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 30 (1997), 4619-4627.  doi: 10.1016/S0362-546X(97)00169-7.

[11]

N. T. Chung, Existence of solutions for nonlocal problems in Orlicz-Sobolev spaces via genus theory, Acta Univ. Apulensis Math. Inform., 37 (2014), 111-123. 

[12]

Ph. ClémentB. de PagterG. Sweers and F. de Thélin, Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces, Mediterr. J. Math., 1 (2004), 241-267.  doi: 10.1007/s00009-004-0014-6.

[13]

Ph. ClémentM. García-HuidobroR. Manásevich and K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations, Calc. Var. Partial Differential Equations, 11 (2000), 33-62.  doi: 10.1007/s005260050002.

[14]

F. J. S. A. Corrêa and G. M. Figueiredo, On a $p$-Kirchhoff equation via Krasnoselskii's genus, Appl. Math. Lett., 22 (2009), 819-822.  doi: 10.1016/j.aml.2008.06.042.

[15]

L. Damascelli and B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of $m$-Laplace equations, J. Differential Equations, 206 (2004), 483-515.  doi: 10.1016/j.jde.2004.05.012.

[16]

G. Dinca, A fixed point method for the $p(\cdot)$-Laplacian, C. R. Math. Acad. Sci. Paris, 347 (2009), 757–762. doi: 10.1016/j.crma.2009.04.022.

[17]

T. K. Donaldson and N. S.Trudinger, Orlicz-Sobolev spaces and imbedding theorems, J. Functional Analysis, 8 (1971), 52-75.  doi: 10.1016/0022-1236(71)90018-8.

[18]

J. Dugundji and A. Granas, Fixed Point Theory, I. Monografie Matematyczne, vol. 61. PWN, Warsaw, 1982.

[19]

M. GarcIa-HuidobroV. K. LeR. Manásevich and K. Schmitt, On principal eigenvalues for quasilinear elliptic differential operators: An Orlicz-Sobolev space setting, Nonlinear Differential Equations Appl., 6 (1999), 207-225.  doi: 10.1007/s000300050073.

[20]

J.-P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc., 190 (1974), 163-205.  doi: 10.1090/S0002-9947-1974-0342854-2.

[21]

J. R. GraefS. Heidarkhani and L. Kong, A variational approach to a Kirchhoff-type problem involving two parameters, Results. Math., 63 (2013), 877-889.  doi: 10.1007/s00025-012-0238-x.

[22]

T. C. Halsey, Electrorheological fluids, Science, 258 (1992), 761-766.  doi: 10.1126/science.258.5083.761.

[23]

G. Kirchhoff, Mechanik, Teubner, Leipzig, (1883).

[24]

M. A. Krasnosel'ski and Ja. B. Ruticki, Convex Functions and Orlicz Spaces, Translated from the first Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen, 1961.

[25]

J. Lamperti, On the isometries of certain function-spaces, Pacific J. Math., 8 (1958), 459-466.  doi: 10.2140/pjm.1958.8.459.

[26]

D. Liu, On a $p$-Kirchhoff equation via fountain theorem and dual fountain theorem, Nonlinear Anal., 72 (2010), 302-308.  doi: 10.1016/j.na.2009.06.052.

[27]

M. Mihǎilescu and V. Rǎdulescu, Neumann problems associated to non-homogeneous differential operators in Orlicz-Sobolev spaces, Ann. Inst. Fourier, 6 (2008), 2087-2111.  doi: 10.5802/aif.2407.

[28]

M. Ruzička, Electrorheological Fluids Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2002.

[29]

I. Samar, Méthodes Variationnelles: Applications á l'analyse d'image et au Modèle de Frenkel-Kontorova, 2011. https://tel.archives-ouvertes.fr/tel-00808646

[30]

E. Zeidler, Nonlinear Functional Analysis and Applications in Nonlinear Monotone Operators, Vol. II/B, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

[31]

V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710.  doi: 10.1070/IM1987v029n01ABEH000958.

show all references

References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. 
[2]

G. A. AfrouziS. Heidarkhani and S. Shokooh, Infinitely many solutions for Steklov problems associated to non-homogeneous differential operators through Orlicz-Sobolev spaces, Complex Var. Elliptic Equ., 60 (2015), 1505-1521.  doi: 10.1080/17476933.2015.1031122.

[3]

C. O. AlvesF. S. J. A. Corrâa and T. F. Ma, Positive solutions for a quasilinear elliptic equations of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93.  doi: 10.1016/j.camwa.2005.01.008.

[4]

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.

[5]

E. AzroulA. Benkirane and M. Srati, Nonlocal eigenvalue type problem in fractional Orlicz-Sobolev space, Adv. Oper. Theory, 5 (2020), 1599-1617.  doi: 10.1007/s43036-020-00067-5.

[6]

E. Azroul, A. Benkirane and M. Shimi, Existence and multiplicity of solutions for fractional $p(x, .)-$Kirchhoff type problems in $\mathbb{R}^N, $, Applicable Analysis, (2019). doi: 10.1080/00036811.2019.1673373.

[7]

E. AzroulA. Benkirane and M. Srati, Existence of solutions for a nonlocal type problem in fractional Orlicz Sobolev spaces, Adv. Oper. Theory, 5 (2020), 1350-1375.  doi: 10.1007/s43036-020-00042-0.

[8]

A. Boumazourh and M. Srati, Leray-Schauder's solution for a nonlocal problem in a fractional Orlicz-Sobolev space, Moroccan J. of Pure and Appl. Anal. (MJPAA), (2020), 42–52.

[9]

F. Cammaroto and L. Vilasi, Multiple solutions for a Kirchhoff-type problem involving the $p(x)$-Laplacian operator, Nonlinear Anal., 74 (2011), 1841-1852.  doi: 10.1016/j.na.2010.10.057.

[10]

M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 30 (1997), 4619-4627.  doi: 10.1016/S0362-546X(97)00169-7.

[11]

N. T. Chung, Existence of solutions for nonlocal problems in Orlicz-Sobolev spaces via genus theory, Acta Univ. Apulensis Math. Inform., 37 (2014), 111-123. 

[12]

Ph. ClémentB. de PagterG. Sweers and F. de Thélin, Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces, Mediterr. J. Math., 1 (2004), 241-267.  doi: 10.1007/s00009-004-0014-6.

[13]

Ph. ClémentM. García-HuidobroR. Manásevich and K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations, Calc. Var. Partial Differential Equations, 11 (2000), 33-62.  doi: 10.1007/s005260050002.

[14]

F. J. S. A. Corrêa and G. M. Figueiredo, On a $p$-Kirchhoff equation via Krasnoselskii's genus, Appl. Math. Lett., 22 (2009), 819-822.  doi: 10.1016/j.aml.2008.06.042.

[15]

L. Damascelli and B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of $m$-Laplace equations, J. Differential Equations, 206 (2004), 483-515.  doi: 10.1016/j.jde.2004.05.012.

[16]

G. Dinca, A fixed point method for the $p(\cdot)$-Laplacian, C. R. Math. Acad. Sci. Paris, 347 (2009), 757–762. doi: 10.1016/j.crma.2009.04.022.

[17]

T. K. Donaldson and N. S.Trudinger, Orlicz-Sobolev spaces and imbedding theorems, J. Functional Analysis, 8 (1971), 52-75.  doi: 10.1016/0022-1236(71)90018-8.

[18]

J. Dugundji and A. Granas, Fixed Point Theory, I. Monografie Matematyczne, vol. 61. PWN, Warsaw, 1982.

[19]

M. GarcIa-HuidobroV. K. LeR. Manásevich and K. Schmitt, On principal eigenvalues for quasilinear elliptic differential operators: An Orlicz-Sobolev space setting, Nonlinear Differential Equations Appl., 6 (1999), 207-225.  doi: 10.1007/s000300050073.

[20]

J.-P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc., 190 (1974), 163-205.  doi: 10.1090/S0002-9947-1974-0342854-2.

[21]

J. R. GraefS. Heidarkhani and L. Kong, A variational approach to a Kirchhoff-type problem involving two parameters, Results. Math., 63 (2013), 877-889.  doi: 10.1007/s00025-012-0238-x.

[22]

T. C. Halsey, Electrorheological fluids, Science, 258 (1992), 761-766.  doi: 10.1126/science.258.5083.761.

[23]

G. Kirchhoff, Mechanik, Teubner, Leipzig, (1883).

[24]

M. A. Krasnosel'ski and Ja. B. Ruticki, Convex Functions and Orlicz Spaces, Translated from the first Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen, 1961.

[25]

J. Lamperti, On the isometries of certain function-spaces, Pacific J. Math., 8 (1958), 459-466.  doi: 10.2140/pjm.1958.8.459.

[26]

D. Liu, On a $p$-Kirchhoff equation via fountain theorem and dual fountain theorem, Nonlinear Anal., 72 (2010), 302-308.  doi: 10.1016/j.na.2009.06.052.

[27]

M. Mihǎilescu and V. Rǎdulescu, Neumann problems associated to non-homogeneous differential operators in Orlicz-Sobolev spaces, Ann. Inst. Fourier, 6 (2008), 2087-2111.  doi: 10.5802/aif.2407.

[28]

M. Ruzička, Electrorheological Fluids Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2002.

[29]

I. Samar, Méthodes Variationnelles: Applications á l'analyse d'image et au Modèle de Frenkel-Kontorova, 2011. https://tel.archives-ouvertes.fr/tel-00808646

[30]

E. Zeidler, Nonlinear Functional Analysis and Applications in Nonlinear Monotone Operators, Vol. II/B, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

[31]

V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710.  doi: 10.1070/IM1987v029n01ABEH000958.

[1]

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

[2]

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

[3]

Jun Wang, Lu Xiao. Existence and concentration of solutions for a Kirchhoff type problem with potentials. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7137-7168. doi: 10.3934/dcds.2016111

[4]

Shu-Zhi Song, Shang-Jie Chen, Chun-Lei Tang. Existence of solutions for Kirchhoff type problems with resonance at higher eigenvalues. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6453-6473. doi: 10.3934/dcds.2016078

[5]

Jiafeng Liao, Peng Zhang, Jiu Liu, Chunlei Tang. Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1959-1974. doi: 10.3934/dcdss.2016080

[6]

Norihisa Ikoma. Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 943-966. doi: 10.3934/dcds.2015.35.943

[7]

Qi-Lin Xie, Xing-Ping Wu, Chun-Lei Tang. Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2773-2786. doi: 10.3934/cpaa.2013.12.2773

[8]

Xiao-Jing Zhong, Chun-Lei Tang. The existence and nonexistence results of ground state nodal solutions for a Kirchhoff type problem. Communications on Pure and Applied Analysis, 2017, 16 (2) : 611-628. doi: 10.3934/cpaa.2017030

[9]

Yongqiang Fu, Xiaoju Zhang. Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1301-1322. doi: 10.3934/dcdsb.2021091

[10]

Igor Chueshov, Irena Lasiecka. Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 777-809. doi: 10.3934/dcds.2006.15.777

[11]

Maria-Magdalena Boureanu. Fourth-order problems with Leray-Lions type operators in variable exponent spaces. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 231-243. doi: 10.3934/dcdss.2019016

[12]

Jeong Ja Bae, Mitsuhiro Nakao. Existence problem for the Kirchhoff type wave equation with a localized weakly nonlinear dissipation in exterior domains. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 731-743. doi: 10.3934/dcds.2004.11.731

[13]

Peng Chen, Xiaochun Liu. Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2018, 17 (1) : 113-125. doi: 10.3934/cpaa.2018007

[14]

Xing Liu, Yijing Sun. Multiple positive solutions for Kirchhoff type problems with singularity. Communications on Pure and Applied Analysis, 2013, 12 (2) : 721-733. doi: 10.3934/cpaa.2013.12.721

[15]

Yuanxiao Li, Ming Mei, Kaijun Zhang. Existence of multiple nontrivial solutions for a $p$-Kirchhoff type elliptic problem involving sign-changing weight functions. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 883-908. doi: 10.3934/dcdsb.2016.21.883

[16]

Wenjing Chen. Multiplicity of solutions for a fractional Kirchhoff type problem. Communications on Pure and Applied Analysis, 2015, 14 (5) : 2009-2020. doi: 10.3934/cpaa.2015.14.2009

[17]

Nemat Nyamoradi, Kaimin Teng. Existence of solutions for a Kirchhoff-type-nonlocal operators of elliptic type. Communications on Pure and Applied Analysis, 2015, 14 (2) : 361-371. doi: 10.3934/cpaa.2015.14.361

[18]

Piotr Gwiazda, Piotr Minakowski, Agnieszka Świerczewska-Gwiazda. On the anisotropic Orlicz spaces applied in the problems of continuum mechanics. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1291-1306. doi: 10.3934/dcdss.2013.6.1291

[19]

Sungwon Cho. Alternative proof for the existence of Green's function. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1307-1314. doi: 10.3934/cpaa.2011.10.1307

[20]

Abdelmalek Aboussoror, Samir Adly, Vincent Jalby. Weak nonlinear bilevel problems: Existence of solutions via reverse convex and convex maximization problems. Journal of Industrial and Management Optimization, 2011, 7 (3) : 559-571. doi: 10.3934/jimo.2011.7.559

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (399)
  • HTML views (456)
  • Cited by (0)

[Back to Top]