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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 |
$ \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*} $ |
$ \Omega $ |
$ {\mathbb{R}}^N $ |
$ N\geq 1 $ |
$ \partial \Omega. $ |
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] | |
[24] |
M. A. Krasnosel'ski |
[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] | |
[24] |
M. A. Krasnosel'ski |
[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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