The paper deals with the study of the existence result for a Kirchhoff elliptic system with additive right hand side and variable parameters involving $ p- $Laplacian operator by using the sub-super solutions method. Our study is an natural extension result of our previous once in (Math. Methods Appl. Sci. 41 (2018), 5203–5210), where in the latter we discussed only the simple case when the parameters are constant.
Citation: Salah Boulaaras, Rafik Guefaifia, Bahri Cherif, Taha Radwan. Existence result for a Kirchhoff elliptic system involving p-Laplacian operator with variable parameters and additive right hand side via sub and super solution methods[J]. AIMS Mathematics, 2021, 6(3): 2315-2329. doi: 10.3934/math.2021140
The paper deals with the study of the existence result for a Kirchhoff elliptic system with additive right hand side and variable parameters involving $ p- $Laplacian operator by using the sub-super solutions method. Our study is an natural extension result of our previous once in (Math. Methods Appl. Sci. 41 (2018), 5203–5210), where in the latter we discussed only the simple case when the parameters are constant.
| [1] | C. O. Alves, F. J. S. A. Correa, On existence of solutions for a class of problem involving a nonlinear operator, Comm. Appl. Nonlinear Anal., 8 (2001), 43–56. |
| [2] |
M. Alizadeh, M. Alimohammady, Regularity and entropy solutions of some elliptic equations, Miskolc Math. Notes, 19 (2018), 715–729. doi: 10.18514/MMN.2018.2545
|
| [3] |
N. Azouz, A. Bensedik, Existence result for an elliptic equation of Kirchhoff-type with changing sign data, Funkcialaj Ekvacioj, 55 (2012), 55–66. doi: 10.1619/fesi.55.55
|
| [4] |
Y. Bouizem, S. Boulaaras, B. Djebbar, Existence of positive solutions for a class of Kirchhof elliptic systems with right hand side defined as a multiplication of two separate functions, Kragujevac J. Math., 45 (2021), 587–596. doi: 10.46793/KgJMat2104.587B
|
| [5] |
Y. Bouizem, S. Boulaaras, B. Djebbar, Some existence results for an elliptic equation of Kirchhoff-type with changing sign data and a logarithmic nonlinearity, Math. Methods Appl. Sci., 42 (2019), 2465–2474. doi: 10.1002/mma.5523
|
| [6] | S. Boulaaras, Existence of positive solutions for a new class of parabolic Kirchoff systems with right hand side defined as a multiplication of two separate functions, Rocky Mt. J. Math., 50 (2020), 445–454. |
| [7] |
F. J. S. A. Correa, G. M. Figueiredo, On a p-Kirchhoff equation type via Krasnoselkii's genus, Appl. Math. Lett., 22 (2009), 819–822. doi: 10.1016/j.aml.2008.06.042
|
| [8] |
S. Gala, Q. Liu, M. A. Ragusa, A new regularity criterion for the nematic liquid crystal fows, Appl. Anal., 91 (2012), 1741–1747. doi: 10.1080/00036811.2011.581233
|
| [9] |
S. Gala, M. A. Ragusa, Logarithmically improved regularity criterion for the Boussinesq equations in Besov spaces with negative indices, Appl. Anal., 95 (2016), 1271–1279. doi: 10.1080/00036811.2015.1061122
|
| [10] | L. C. Evans, Partial Differential Equations, 2nd Edition, Berkeley, American Mathematical Society, 2010. |
| [11] | X. L. Fan, D. Zhao, On the spaces $L^{p\left(x\right) }\left(\Omega \right) $ and $W^{m, p\left(x\right) }\left(\Omega \right) $, J. Math. Anal. Appl., 263 (2001), 424–446. |
| [12] | S. Boulaaras, Existence of positive solutions of nonlocal p(x)-Kirchhoff hyperbolic systems via sub-super solutions concept, J. Intell. Fuzzy Syst., 38 (2020), 1–13. |
| [13] |
G. M. Figueiredo, A. Suarez, Some remarks on the comparison principle in Kirchhof equations, Rev. Mat. Iberoam., 34 (2018), 609–620. doi: 10.4171/RMI/997
|
| [14] |
J. Garcia-Melian, L. Iturriaga, Some counter examples related to the stationary Kirchhof equation, Proc. Amer. Math. Soc., 144 (2016), 3405–3411. doi: 10.1090/proc/12971
|
| [15] | G. R. Kirchhoff, Vorlesungen über mathematische Physik-Mechanik, 3 Edition, Teubner, Leipzig, 1883. |
| [16] |
D. Lu, S. Peng, Existence and asymptotic behavior of vector solutions for coupled nonlinear Kirchhoff-type systems, J. Differ. Equations, 263 (2017), 8947–8978. doi: 10.1016/j.jde.2017.08.062
|
| [17] |
D. Lu, J. Xiao, Ground state solutions for a coupled Kirchhoff-type system, Math. Methods Appl. Sci., 38 (2015), 4931–4948. doi: 10.1002/mma.3414
|
| [18] | D. Lu, Existence and multiplicity results for perturbed Kirchhoff-type Schrödinger systems in $R^{3}$, Comput. Math. Appl., 68 (2014), 1180–1193. |
| [19] |
T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), 1967–1977. doi: 10.1016/j.na.2005.03.021
|
| [20] | S. Polidoro, M. A. Ragusa, Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term, Rev. Mat. Iberoam., 24 (2008), 1011–1046. |
| [21] |
B. Ricceri, On an elliptic Kirchhoff-type problem depending on two parameters, J. Global Optim., 46 (2010), 543–549. doi: 10.1007/s10898-009-9438-7
|
| [22] |
S. Boulaaras, R. Guefaifia, Existence of positive weak solutions for a class of Kirrchoff elliptic systems with multiple parameters, Math. Methods Appl. Sci., 41 (2018), 5203–5210. doi: 10.1002/mma.5071
|
| [23] |
J. J. Sun, C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal. Theory Methods Appl., 74 (2011), 1212–1222. doi: 10.1016/j.na.2010.09.061
|
| [24] | G. A. Afrouzi, N. T. Chung, S. Shakeri, Existence of positive solutions for kirchhoff type equations, Electron. J. Differ. Equations, 2013 (2013), 1–8. |
Salah Boulaaras, Rafik Guefaifia, Bahri Cherif, Taha Radwan. Existence result for a Kirchhoff elliptic system involving p-Laplacian operator with variable parameters and additive right hand side via sub and super solution methods[J]. AIMS Mathematics, 2021, 6(3): 2315-2329. doi: 10.3934/math.2021140
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