EXISTENCE OF SOLUTIONS FOR A DEGENERATE QUASILINEAR ELLIPTIC SYSTEM IN BOUNDED DOMAIN

G. A. Afrouzi; N. T. Chung; M. Mirzapour

doi:10.11948/2013001

2013 Volume 3 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2013 > 3(1): 1-9

Next Article Previous Article

G. A. Afrouzi, N. T. Chung, M. Mirzapour. EXISTENCE OF SOLUTIONS FOR A DEGENERATE QUASILINEAR ELLIPTIC SYSTEM IN BOUNDED DOMAIN[J]. Journal of Applied Analysis & Computation, 2013, 3(1): 1-9. doi: 10.11948/2013001

Citation:

G. A. Afrouzi, N. T. Chung, M. Mirzapour. EXISTENCE OF SOLUTIONS FOR A DEGENERATE QUASILINEAR ELLIPTIC SYSTEM IN BOUNDED DOMAIN[J]. Journal of Applied Analysis & Computation, 2013, 3(1): 1-9. doi: 10.11948/2013001

EXISTENCE OF SOLUTIONS FOR A DEGENERATE QUASILINEAR ELLIPTIC SYSTEM IN BOUNDED DOMAIN

1 Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran;
2 Department of Mathematics and Informatics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Vietnam

Abstract

Abstract

Using variational methods, we study the existence of weak solutions for the degenerate quasilinear elliptic system { -div(h₁(x)|∇u|^p-2∇u)=F_u(x, u, v) in Ω, -div(h₂(x)|∇u|^p-2∇u)=F_u(x, u, v) in Ω, u=v=0 on ∂Ω, where Ω ⊂ R^N is a smooth bounded domain, ∇F=(F_u, F_v) stands for the gradient of C¹-function F:Ω×R² → R, the weights h_i, i=1, 2 are allowed to vanish somewhere, the primitive F(x, u, v) is intimately related to the first eigenvalue of a corresponding quasilinear system.
- Quasilinear degenerate elliptic system /
- Palais-Smale condition /
- mountain pass theorem /
- existence
MSC: 35J60;35B30;35B40

FullText(HTML)

References

[1]	G.A. Afrouzi, S. Mahdavi and Nikolaos B. Zographopoulos, Existence of solutions for non-uniformly nonlinear elliptic systems, Electron. J. Diff. Equ., 167(2011), 1-9. Google Scholar
[2]	G.A. Afrouzi and S. Mahdavi, Existence results for a class of degenerate quasilinear elliptic systems, Lithuanian Math. J., 51(2011), 451-460. Google Scholar
[3]	A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal., 4(1973), 349-381. Google Scholar
[4]	L. Boccardo and D.G. De Figueiredo, Some remarks on a system of quasilinear elliptic equations, Nonlinear Diff. Equ. Appl. (NoDEA), 9(2002), 309-323. Google Scholar
[5]	P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, Nonlinear Diff. Equ. Appl. (NoDEA), 7(2000), 187-199. Google Scholar
[6]	N.T. Chung, Existence of infinitely many solutions for degenerate and singular elliptic systems with indefinite concave nonlinearities, Electron. J. Diff. Equ., 30(2011), 1-12. Google Scholar
[7]	N.T. Chung and H.Q. Toan, On a class of degenerate and singular elliptic systems in bounded domain, J. Math. Anal. Appl., 360(2009), 422-431. Google Scholar
[8]	D.G. Costa, On a class of elliptic systems in R^N, Electron. J. Diff. Equ., 7(1994), 1-14. Google Scholar
[9]	A. Djellit and S. Tas, Existence of solutions for a class of elliptic systems in R^N involving the p-Laplacian, Electron. J. Diff. Equ., 56(2003), 1-8. Google Scholar
[10]	P. Drabek, A. Kufner and F. Nicolosi, Quasilinear elliptic equations with degeneration and singularities, Walter de Gruyter and Co., Berlin, 1997. Google Scholar
[11]	D.D. Hai and R. Shivaji, An existence result on positive solutions of p-Laplacian systems, Nonlinear Anal., 56(2004), 1007-1010. Google Scholar
[12]	D.W. Huang and Y.Q. Li, Multiplicity of solutions for a noncooperative pLaplacian elliptic system in R^N, J. Differential Equations, 215(2005), 206-223. Google Scholar
[13]	S. Ma, Nontrivial solutions for resonant cooperative elliptic systems via computations of the critical groups, Nonlinear Analysis, 73(2010), 3856-3872. Google Scholar
[14]	J.M.B. do Ó, Solutions to perturbed eigenvalue problems of the p-Laplacian in RN, Electron. J. Diff. Equ., 11(1997), 1-15. Google Scholar
[15]	T.F. Wu, The Nehari manifold for a semilinear elliptic system involving signchanging weight functions, Nonlinear Analysis, 68(2008), 1733-1745. Google Scholar
[16]	G. Zhang and Y. Wang, Some existence results for a class of degenerate semilinear elliptic systems, J. Math. Anal. Appl., 333(2007), 904-918. Google Scholar
[17]	N.B. Zographopoulos, On the principal eigenvalue of degenerate quasilinear elliptic systems, Math. Nachr., 281(2008), No. 9, 1351-1365. Google Scholar
[18]	N.B. Zographopoulos, On a class of degenarate potential elliptic system, Nonlinear Diff. Equ. Appl. (NoDEA), 11(2004), 191-199. Google Scholar
[19]	N.B. Zographopoulos, p-Laplacian systems on resonance, Appl. Anal., 83(2004), 509-519. Google Scholar