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

Abdelaaziz Sbai; Youssef El Hadfi; Mohammed Srati; Noureddine Aboutabit

doi:10.3934/dcdss.2021015

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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 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. 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.
[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.
[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. 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.
[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. 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.
[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. 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.
[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.
[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-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.
[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. 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.
[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. 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.
[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.
[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. 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.
[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. 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.
[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. 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.
[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.
[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-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.
[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. 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.
[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.

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American Institute of Mathematical Sciences

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

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

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