Positive Solution for a Class of Infinite Semipositone (p,q)-Laplace System
DOI:
https://doi.org/10.5890/DNC.2022.12.013Abstract
In this paper we consider following (p,q)-Laplacian system $$ \left\{ \begin{aligned} & -\Delta _{p}u=\lambda l\left( x\right) u^{p-1}-f_{1}\left( u,v\right) -au^{-\alpha _{1}}v^{\beta _{2}}\ \text{in }\Omega , \\ & -\Delta _{q}v=\mu k\left( x\right) v^{q-1}-f_{2}\left( u,v\right) -bu^{\alpha _{2}}v^{-\beta _{2}}\text{ in }\Omega , \\ & u=v=0\text{ on }\partial \Omega , \end{aligned} \right. $$ where $\Omega $ is a bounded domain in $\mathbb{R}^{N}$ with smooth boundary $\partial \Omega $, $\lambda $ and $\mu $ are a positive parameters and $a,$ $b$ are a positive constant. By using the method of sub-supersolution we discuss the existence of positive solution.References
[1] Boulaaras, S. and Guefaifi, R. (2018), Existence of positive solutions for a class of -Laplacian systems (p(x), q (x)), Rendiconti del Circolo Matematico di Palermo, 2(67), 93-103.
[2] Boulaaras, S.G. and Bouali, T. (2018), Existence de solutions positives pour une classe de syst`{e}mes elliptiques singuliers quasi-lin{e}aires impliquant un exposant de Caffarelli-Kohn-Nirenberg avec des fonctions de poids de changement de signe, Indian Journal of Pure and Applied Mathematics, 2018.
[3] Brezis, H. (1987), Analyse fonctionnelle, th{e}orie et applications, Masson. Parie.
[4] Demengel, F. and Demengel, G. (2007), Espaces fonctionnels. utilisation dans des E.D.P, Sciences France.
[5] Guefaifia, R. and Boulaaras (2020), Sub-super solutions method for elliptical systems hold Laplacian p 1, ..., pm, Mathematical Methods in Applied Sciences, 43(7), 4191-4199.
[6] Guefaifia, R., Akrout, K., and Saifia, W. (2013), Existence and nonexistence of weak positive solution for classes of 3 p-Laplacian elliptic systems, International Journal of Partial Differential Equations and Applications, 1(1), 13-17.
[7] Guefaifia, R., Boulaaras, S.M., Alodhaibi, S., and Alkhalaf, S. (2020), Existence of weak positive solutions for a new class of Laplacian nonlinear elliptical system with sign change weights, Complexity, 2020.
[8] Guefaifia, R., Boulaaras, S., and Bouizem, Y. (2020), Existence of positive solutions for a class of Kirrchoff systems with the right side defined as a multiplication of two distinct functions, Applied Mathematics E-Notes, 19, 331-342.
[9] Guefaifia, R., Zuo, J., Boulaaras, S., and Agarwal, P. (2020), Existence and multiplicity of positive weak solutions for a new class of (p; q) -laplacian systems preprint arXiv arXiv: 2006.05776.
[10] Haghaieghi, S. and Afrouzi, G.A. (2011), Sub-super solutions for (p-q) laplacian systems, Boundary Value Problems, 52.
[11] Hai, D.D. and Shivaji, R. (2004), An existence result on positive solutions for a class of p-laplacian systems, Nonlinear Analysis, 56, 1007-1010.
[12] Lee, E.K., Shivaji, R., and Ye. J, (2010), Positive solutions for infinite semipositone problems with falling zeros, Nonlinear Analysis, 72, 4475-4479.
[13] Rasouli, S.H. (2013), On the existence of positive solutions for a class of infinite semipositone systems with singular weights, Thai J. Math., 11(1), 103-110.
[14] Shivaji, R. and Ye, J. (2011), Nonexistence results for classes of elliptic systems, Nonlinear Analysis, 74, 1485-1494. 189-204.
[15] Zediri, S., Guefaifia, R., and Boulaaras, S. (2020), Existence of positive solutions of a new classof nonlocal p(x)-Kirchhoff parabolic systemsvia sub-super-solutions concept, J. Appl. Anal., 26(1), 49-58.
[16] Guefaifia, R., Boulaaras, S., Cherif, B., and Radwan, T. (2020), Infinite existence solutions of fractional systems with lipschitz nonlinearity, Journal of Function Spaces, vol. 2020, Article ID 6679101, 11 pages, 2020. https://doi.org/ 10.1155/2020/6679101
[17] Boulaaras, S., Guefaifia, R., Cherif, B., and Radwan, T. (2020), 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, AIMS Mathematics, 6(3), 2315-2329. doi: 10.3934/math.2021140
[18] Guefaifia, R., Boulaaras, J., and Zuo, P.A. (2020), Existence and multiplicity of positive weak solutions for a new class of (P; Q)-Laplacian systems, Miskolc Mathematical Notes, 21(2), 861-872.
[19] Haiour, M., Boulaaras, S., Guefaifia, R., and Kamache, F. (2020), Existence result for a new class of Kirchhoff elliptic system with variable parameters, Miskolc Mathematical Notes, 21(2), 887-896.
[20] Afrouzi, G.A., Shakeri, S., and Chung, N.T. (2013), Remark on an infinite semipositone problem with indefinite weight and falling zeros, Proc. Indian Acad. Sci. (Math. Sci.), 123(1), 145-150.
[21] Akrout, K. (2015), Existence of positive solution for a class of infinite semipositone p-Laplace systems with falling zeros, International Journal of Mathematics and Computation, 26(4), 74-80.
[22] Afrouzi, G.A. Chung, N.T., and Shakeri, S. (2013), Existence of positive solutions for kirchhoff type equations, Electronic Journal of Differential Equations, 2013(180), 1-8.
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