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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

Abdelaaziz Sbai 1, , Youssef El Hadfi 1,, , Mohammed Srati 2, and  Noureddine Aboutabit 1,

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.
Keywords: Existence, Leray Schauder's alternative theorem, Orlicz-Sobolev spaces, nonlinear problems, weak solutions, Kirchhoff type problem.
Mathematics Subject Classification: Primary: 35J20, 35J60, 47G20, 49J40.
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 & 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.   Google Scholar
[2]

G. A. Afrouzi, S. 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.  Google Scholar

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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

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E. Azroul, A. 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.  Google Scholar

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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.  Google Scholar

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E. Azroul, A. 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.  Google Scholar

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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. Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.   Google Scholar

[12]

Ph. Clément, B. de Pagter, G. 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.  Google Scholar

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Ph. Clément, M. García-Huidobro, R. 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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

M. GarcIa-Huidobro, V. K. Le, R. 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.  Google Scholar

[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.  Google Scholar

[21]

J. R. Graef, S. 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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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 Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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

G. A. Afrouzi, S. 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.  Google Scholar

[3]

C. O. Alves, F. 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.  Google Scholar

[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.  Google Scholar

[5]

E. Azroul, A. 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.  Google Scholar

[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.  Google Scholar

[7]

E. Azroul, A. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[12]

Ph. Clément, B. de Pagter, G. 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.  Google Scholar

[13]

Ph. Clément, M. García-Huidobro, R. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

M. GarcIa-Huidobro, V. K. Le, R. 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.  Google Scholar

[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.  Google Scholar

[21]

J. R. Graef, S. 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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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 Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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