Maximum Principle and Existence of Weak Solutions for Nonlinear System Involving Singular (p,q)-Laplacian Operators
Keywords:
Maximum principle, weak solution, singular p-Laplacian.
Abstract
We investigate in this work necessary and sufficient conditions for having the maximum principle for nonlinear system involving singular (p,q)-Laplacian operators on bounded domain Ω of Rn. Moreover, we prove the existence of positive weak solutions by the Browder theorem method for the considered system
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References
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[2] W. Bastos, O. Miyagaki and R. Vieira, Existence of solutions for a class of degenerate quasilinear elliptic equation in 𝑅^𝑛 with vanishing potentials, Boundary Value Problems, 92 (2013), 1.16
[3] M. Bouchekif, H. Serag and F. de Thélin, On Maximum Principle and Existence of Solutions for Some Nonlinear Elliptic Systems, Rev. Mat. Apl., 16 (1995) 1.16
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[8] S. Khafagy and H. Serag, Maximum principle and existence of weak solutions for weighted p-Laplacian system on 𝑅𝑛, Asian Journal of Mathematics and Computer Research, 6 2 (2015) 168-180.
[9] S. Khafagy and M. Herzallah, Maximum Principle and Existence of Weak Solutions for Nonlinear System Involving Weighted (p, q)-Laplacian, to appear in Southeast Asian Bulletin of Mathematics.
[10] L. Leadi and A. Marcos, Maximum Principle and Existence Results for Nonlinear Cooperative Systems on a Bounded Domain, Electron. J. Diff. Eqns., 2011 82 (2011) 1.13
[11] J. Leray and J. Lions, Quelques resultats de Visik sur les proble.mes elliptiques nonlineaires par les methodes de Minty Browder. Bull. Soc. Math. France. 107 (1995) 93.97
[12] O. Miyagaki, R. Rodrigues, On positive solutions for a class of singular quasilinear elliptic systems. J. Math. Anal. Appl. 334 (2007) 818.833
[13] S. Rasouli, On a class of singular elliptic system with combined nonlinear effects, Acta Universitatis Apulensis, 38 (2014) 187.195
[14] H. Serag and E. El-Zahrani E, Maximum Principle and Existence of Positive Solution for Nonlinear Systems on 𝑅^𝑛, Electron. J. Di . Eqns., 85 (2005) 1-12
[15] Z. Xiy and C. Chen, Existence of multiple solutions for singular elliptic problems with nonlinear boundary conditions, J. Math. Anal. Appl., 410 (2014) 625.641
[16] B. Xuan, Multiple solutions to a Caffarelli-Kohn-Nirenberg type equation with asymptotically linear term, Revista Colombiana de Matematicas, 37 (2003) 65.79
[17] B. Xuan, The eigenvalue problem for a singular quasilinear elliptic equation, Electron. J. Diff. Eqns., 2004 16 (2004) 1.11.
[18] B. Xuan, The solvability of quasilinear Brezis-Nirenberg-type problems with singular weights. Nonlinear Anal. 62 (2005) 703.725.
[2] W. Bastos, O. Miyagaki and R. Vieira, Existence of solutions for a class of degenerate quasilinear elliptic equation in 𝑅^𝑛 with vanishing potentials, Boundary Value Problems, 92 (2013), 1.16
[3] M. Bouchekif, H. Serag and F. de Thélin, On Maximum Principle and Existence of Solutions for Some Nonlinear Elliptic Systems, Rev. Mat. Apl., 16 (1995) 1.16
[4] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights. Compos. Math. 53 (1984) 259.275
[5] F. Catrina, Z-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions. Commun. Pure Appl. Math. 54 2 (2201) 229.258
[6] J. Fleckinger and H. Serag, On Maximum Principle and Existence of Solutions for Elliptic Systems on 𝑅𝑛, J. Egypt. Math. Soc. 2 (1994) 45-51.
[7] J. Francu, .Solvability of Operator Equations., Survey Directed to Differential Equations, Lecture Notes of IMAMM 94, Proc. of the Seminar . Industrial Mathematics and Mathematical Modelling., Rybnik, Univ. West Bohemia in Pilsen, Faculty of Applied Sciences, Dept. of Math., July, 4.8, 1994.
[8] S. Khafagy and H. Serag, Maximum principle and existence of weak solutions for weighted p-Laplacian system on 𝑅𝑛, Asian Journal of Mathematics and Computer Research, 6 2 (2015) 168-180.
[9] S. Khafagy and M. Herzallah, Maximum Principle and Existence of Weak Solutions for Nonlinear System Involving Weighted (p, q)-Laplacian, to appear in Southeast Asian Bulletin of Mathematics.
[10] L. Leadi and A. Marcos, Maximum Principle and Existence Results for Nonlinear Cooperative Systems on a Bounded Domain, Electron. J. Diff. Eqns., 2011 82 (2011) 1.13
[11] J. Leray and J. Lions, Quelques resultats de Visik sur les proble.mes elliptiques nonlineaires par les methodes de Minty Browder. Bull. Soc. Math. France. 107 (1995) 93.97
[12] O. Miyagaki, R. Rodrigues, On positive solutions for a class of singular quasilinear elliptic systems. J. Math. Anal. Appl. 334 (2007) 818.833
[13] S. Rasouli, On a class of singular elliptic system with combined nonlinear effects, Acta Universitatis Apulensis, 38 (2014) 187.195
[14] H. Serag and E. El-Zahrani E, Maximum Principle and Existence of Positive Solution for Nonlinear Systems on 𝑅^𝑛, Electron. J. Di . Eqns., 85 (2005) 1-12
[15] Z. Xiy and C. Chen, Existence of multiple solutions for singular elliptic problems with nonlinear boundary conditions, J. Math. Anal. Appl., 410 (2014) 625.641
[16] B. Xuan, Multiple solutions to a Caffarelli-Kohn-Nirenberg type equation with asymptotically linear term, Revista Colombiana de Matematicas, 37 (2003) 65.79
[17] B. Xuan, The eigenvalue problem for a singular quasilinear elliptic equation, Electron. J. Diff. Eqns., 2004 16 (2004) 1.11.
[18] B. Xuan, The solvability of quasilinear Brezis-Nirenberg-type problems with singular weights. Nonlinear Anal. 62 (2005) 703.725.
Published
2016-06-06
How to Cite
Khafagy, S. (2016). Maximum Principle and Existence of Weak Solutions for Nonlinear System Involving Singular (p,q)-Laplacian Operators. Journal of Progressive Research in Mathematics, 7(4), 1153-1163. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/773
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