Existence of positive solutions of Kirchhoff hyperbolic systems with multiple parameters
DOI:
https://doi.org/10.5269/bspm.45418Abstract
In this paper, by using sub-super solutions method, we study the existence of weak positive solution of Kirrchoff hyperbolic systems in bounded domains with multiple parameters. These results extend and improve many results in the literature.
References
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17. Kirchhoff , G., Mechanik, Teubner, Leipzig, Germany, 1883.
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20. Boulaaras, S.; Draifia, A.; Alnegga, M. Polynomial Decay Rate for Kirchhoff Type in Viscoelasticity with Logarithmic Nonlinearity and Not Necessarily Decreasing Kernel. Symmetry 2019, 11, 226. https://doi.org/10.3390/sym11020226
21. X. L. Fan and D. Zhao, On the spaces L p(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl., 263 (2001), 424- 446. https://doi.org/10.1006/jmaa.2000.7617
22. Boulaaras, S.; Allahem, A. Existence of Positive Solutions of Nonlocal p(x)-Kirchhoff Evolutionary Systems via SubSuper Solutions Concept. Symmetry 2019, 11, 253. https://doi.org/10.3390/sym11020253
23. X. L. Fan and D. Zhao, The quasi-minimizer of integral functionals with m(x) growth conditions, Nonlinear Anal., 39 (2000), 807-816. https://doi.org/10.1016/S0362-546X(98)00239-9
24. X. L. Fan and D. Zhao, Regularity of minimizers of variational integrals with continuous p(x)−growth conditions, Chinese Ann. Math., 17A (5) (1996), 557-564.
25. X. Han and G. Dai, On the sub-supersolution method for p(x)−Kirchhoff type equations, Journal of Inequalities and Applications, 2012 (2012): 283. https://doi.org/10.1186/1029-242X-2012-283
26. G. Kirchhoff, Mechanik, Teubner, Leipzig, Germany, 1883.
27. T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005),1967-1977. https://doi.org/10.1016/j.na.2005.03.021
28. B. Ricceri, On an elliptic Kirchhoff-type problem depending on two parameters, J. Global Optimization, 46(4) 2010, 543-549. https://doi.org/10.1007/s10898-009-9438-7
29. M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2002.
30. V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR., Izv29, (1987), 33-36. https://doi.org/10.1070/IM1987v029n01ABEH000958
2. Azouz, N, and Bensedik, A., Existence result for an elliptic equation of Kirchhoff -type with changing sign data, Funkcial. Ekvac., 55 (2012), 55-66. https://doi.org/10.1619/fesi.55.55
3. S. Boulaaras, R.Guefaifia, Existence of positive weak solutions for a class of Kirrchoff elliptic systems with multiple parameters, Math Meth Appl Sci., Volume 41, Issue 13, 5203-5210 https://doi.org/10.1002/mma.5071
4. S. Boulaaras, R.Guefaifia and S. Kabli: An asymptotic behavior of positive solutions for a new class of elliptic systems involving of (p(x), q(x))-Laplacian systems. Bol. Soc. Mat. Mex. (2017). https://doi.org/10.1007/s40590-017-0184-4
5. S. Boulaaras, K. Habita and M. Haiour, A posteriori error estimates for the generalized overlapping domain decomposition method for a parabolic variational equation with mixed boundary condition, Bol. Soc. Paran. Mat. v. 38 4 (2020): 111-126. https://doi.org/10.5269/bspm.v38i4.40535
6. S. Boulaaras, B. C. Bahi and M. Haiour, The maximum norm analysis of a nonmatching grids method for a class of parabolic equation with nonlinear source terms, Bol. Soc. Paran. Mat. 38 4 (2020): 157-174. https://doi.org/10.5269/bspm.v38i4.40272
7. Y. Bouizm, S. Boulaaras and B. Djebbar, Some existence results for an elliptic equation of Kirchhoff-type with changing sign data and a logarithmic nonlinearity, Math Meth Appl Sci., (2019), https://doi.org/10.1002/mma.5523
8. Boulaaras, S; Guefaifia, R.; Bouali, T. Existence of positive solutions for a class of quasilinear singular elliptic systems involving Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions. Indian J. Pure Appl. Math. 2018, 49, 705-715. https://doi.org/10.1007/s13226-018-0296-1
9. Chipot, M. and Lovat, B., Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 30 (1997), 4619-4627. https://doi.org/10.1016/S0362-546X(97)00169-7
10. Correa, F. J. S. A. and Figueiredo, G. M., On an elliptic equation of p−Kirchhoff type via variational methods, Bull. Austral. Math. Soc., 74 (2006), 263-277. https://doi.org/10.1017/S000497270003570X
11. Correa, F. J. S. A. and Figueiredo, G. M., On a p−Kirchhoff equation type via Krasnoselkii's genus, Appl. Math. Lett. , 22 (2009), 819-822. https://doi.org/10.1016/j.aml.2008.06.042
12. Hai, D. D. and Shivaji, R., An existence result on positive solutions for a class of p−Laplacian systems, Nonlinear Anal., 56 (2004), 1007-1010. https://doi.org/10.1016/j.na.2003.10.024
13. R. Guefaifia and S. Boulaaras Existence of positive radial solutions for (p(x),q(x))-Laplacian systems Appl. Math. E-Notes, 18(2018), 209-218
14. R. Guefaifia and S. Boulaaras, Existence of positive solution for a class of (p(x),q(x))-Laplacian systems, Rend. Circ. Mat. Palermo, II. Ser 67 (2018), 93-103 https://doi.org/10.1007/s12215-017-0297-7
15. Han, X. and Dai, G., On the sub-supersolution method for p (x) −Kirchhoff type equations, J. Inequal. Appl., 2012: 283 (2012) 11pp. https://doi.org/10.1186/1029-242X-2012-283
16. Medekhel, H.; Boulaaras, S.; Zennir, K.; Allahem, A. Existence of Positive Solutions and Its Asymptotic Behavior of (p(x), q(x))-Laplacian Parabolic System. Symmetry 2019, 11, 332. https://doi.org/10.3390/sym11030332
17. Kirchhoff , G., Mechanik, Teubner, Leipzig, Germany, 1883.
18. Ma, T. F., Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), 1967-1977. https://doi.org/10.1016/j.na.2005.03.021
19. Ricceri, B., On an elliptic Kirchhoff -type problem depending on two parameters, J. Global Optim., 46 (2010), 543-549. https://doi.org/10.1007/s10898-009-9438-7
20. Boulaaras, S.; Draifia, A.; Alnegga, M. Polynomial Decay Rate for Kirchhoff Type in Viscoelasticity with Logarithmic Nonlinearity and Not Necessarily Decreasing Kernel. Symmetry 2019, 11, 226. https://doi.org/10.3390/sym11020226
21. X. L. Fan and D. Zhao, On the spaces L p(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl., 263 (2001), 424- 446. https://doi.org/10.1006/jmaa.2000.7617
22. Boulaaras, S.; Allahem, A. Existence of Positive Solutions of Nonlocal p(x)-Kirchhoff Evolutionary Systems via SubSuper Solutions Concept. Symmetry 2019, 11, 253. https://doi.org/10.3390/sym11020253
23. X. L. Fan and D. Zhao, The quasi-minimizer of integral functionals with m(x) growth conditions, Nonlinear Anal., 39 (2000), 807-816. https://doi.org/10.1016/S0362-546X(98)00239-9
24. X. L. Fan and D. Zhao, Regularity of minimizers of variational integrals with continuous p(x)−growth conditions, Chinese Ann. Math., 17A (5) (1996), 557-564.
25. X. Han and G. Dai, On the sub-supersolution method for p(x)−Kirchhoff type equations, Journal of Inequalities and Applications, 2012 (2012): 283. https://doi.org/10.1186/1029-242X-2012-283
26. G. Kirchhoff, Mechanik, Teubner, Leipzig, Germany, 1883.
27. T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005),1967-1977. https://doi.org/10.1016/j.na.2005.03.021
28. B. Ricceri, On an elliptic Kirchhoff-type problem depending on two parameters, J. Global Optimization, 46(4) 2010, 543-549. https://doi.org/10.1007/s10898-009-9438-7
29. M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2002.
30. V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR., Izv29, (1987), 33-36. https://doi.org/10.1070/IM1987v029n01ABEH000958
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2021-12-20
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