Document Type : Research Paper

Authors

Department of Basic Sciences, Payame Noor University of Karaj, Karaj, Iran.

Abstract

In this paper, we study the multiplicity of positive solutions for the Laplacian systems with sign-changing weight functions. Using the decomposition of the Nehari manifold, we prove that an elliptic system has at least two positive solutions.

Keywords

[1] A. Ambrosetti, H. Brezis and G. Cerami, Combined e ects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994) 519-543.
[2] K. J. Brown and T. F. Wu, A brering map approach to a semilinear elliptic boundary value problem, J. Di . Equns 69 (2007) 1-9.
[3] K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign- changing weight function, J. Di . Equns 193 (2003) 481-499.
[4] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 17 (1947) 324-353.
[5] D. G. de Figueiredo, J. P. Gossez and P. Ubilla, Local superlinearity and sub-linearity for inde nite semilinear elliptic problems, J. Funct. Anal. 199 (2003) 452-467.
[6] N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967) 721-747.
[7] M. Willem, Minimax Theorems, Bikhauser, Boston, (1996).
[8] T. F. Wu, On semilinear elliptic equations involving concave-convex nonlineari-ties ans sign-changing weight function, J. Math. Anal. Appl. 318 (2006) 253-270.
[9] T. F. Wu, Multiplicity results for a semilinear elliptic equation involving sign-changing weight function, Rocky Mountain Journal of Mathematics, 39(2009), 995-1011.
[10] T. F. Wu, Multiplicity of positive solutions of p-Laplacian problems with sign-changing weight functions, J. Math. Anal. Appl. 12 (2007) 557-563.
[11] T. F. Wu, The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions, Nonlinear Anal. 68(2007) 1733-1745.