Abstract
We present a note on the paper by Brown and Wu (J Math Anal Appl 337:1326–1336, 2008). Indeed, we extend the multiplicity results for a class of semilinear elliptic system to the quasilinear elliptic system of the form:
Here Δ p denotes the p-Laplacian operator defined by \({\Delta_{p}z=div\,(|\nabla z|^{p-2} \nabla z), p >2 , \Omega \subset \mathbb{R}^N}\) is a bounded domain with smooth boundary, \({\alpha >1 , \beta >1 , 2 < \alpha+\beta < p < \gamma < p* (p* = \frac{pN}{N-p} {\rm if} N > p, p*=\infty {\rm if} N \,\leq p), \frac{\partial}{\partial n}}\) is the outer normal derivative, \({(\lambda, \mu) \in \mathbb{R}^{2} {\setminus} \{(0,0)\},}\) the weight m(x) is a positive bounded function, and \({a(x), b(x)\in C(\partial \Omega)}\) are functions which change sign in \({\overline{\Omega}.}\)
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adriouch K., El Hamidi A.: The Nehari manifold for systems of nonlinear elliptic equations. Nonlinear Anal. 64(5), 2149–2167 (2006)
Afrouzi G.A., Rasouli S.H.: A remark on the existence and multiplicity result for a nonlinear elliptic problem involving the p-Laplacian. Nonlinear Differ. Equ. Appl. 16, 17–730 (2009)
Alves C.O., El Hamidi A.: Nehari manifold and existence of positive solutions to a class of quasilinear problems. Nonlinear Anal. 60, 611–624 (2005)
Ambrosetti A., Brezis H., Cerami G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122, 519–543 (1994)
Amman H., Lopez-Gomez J.: A priori bounds and multiple solution for superlinear indefinite elliptic problems. J. Differ. Equ. 146, 336–374 (1998)
Atkinson C., El Kalli K.: Some boundary value problems for the Bingham model. J. Non-Newtonian Fluid Mech. 41, 339–363 (1992)
Binding P.A., Drabek P., Huang Y.X.: On Neuman boundary value problems for some quasilinear equations. Nonlinear Anal. 42, 613–629 (2000)
Bouchekif M., Nasri Y.: On a nonhomogeneous elliptic system with changing sign data. Nonlinear Anal. 65, 1476–1487 (2006)
Bozhkov Y., Mitidieri E.: Existence of multiple solutions for quasilinear systems via fibering method. J. Differ. Equ. 190, 239–267 (2003)
Brezis H., Nirenberg L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure. Appl. Math. 36, 437–477 (1983)
Brown K.J.: The Nehari manifold for a semilinear elliptic equation involving a sublinear term. Calc. Var. 22, 483–494 (2005)
Brown K.J., Wu T.F.: A fibering map approach to a semilinear elliptic boundary value problem. Electron. J. Differ. Equ. 69, 1–9 (2007)
Brown K.J., Wu T.F.: A semilinear elliptic system involving nonlinear boundary condition and sign changing weight function. J. Math. Anal. Appl. 337, 1326–1336 (2008)
Brown K.J., Zhang Y.: The Nehari manifold for a semilinear elliptic problem with a sign changing weight function. J. Differ. Equ. 193, 481–499 (2003)
Clément Ph., Fleckinger J., Mitidieri E., de Thelin F.: Existence of positive solutions for quasilinear elliptic systems. J. Differ. Equ. 166(2), 455–477 (2000)
Dancer E.N.: Competing species systems with diffusion and large interaction. Rend. Semin. Mat. Fis. Milano. 65, 23–33 (1995)
Drabek P., Pohozaev S.I.: Positive solutions for the p-Laplacian: application of the fibering method. Proc. R. Soc. Edinburgh Sect A. 127, 721–747 (1997)
Escobar J.F.: Uniqueness theorems on conformal deformations of metrics, Sobolev inequalities, and an eigenvalue estimate. Commun. Pure Appl. Math. 43, 857–883 (1990)
Ladde G.S., Lakshmikantham V., Vatsale A.S.: Existence of coupled quasisolutions of systems of nonlinear elliptic boundary value problems. Nonlinear Anal. 8(5), 501–515 (1984)
Tehrani H.T.: A multiplicity result for the jumping nonlinearity problem. J. Differ. Equ. 118(1), 472–305 (2003)
Tehrani H.T.: On indefinite superlinear elliptic equations. Calc. Var. 4, 139–153 (1996)
Tolksdorf P.: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51, 126–150 (1984)
Wu, T.F.: The Nehari manifold for a semilinear elliptic system involving sign-changing weight function. Nonlinear Anal. (2012) (in press)
Wu T.F.: On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function. J. Math. Anal. Appl. 318, 253–270 (2006)
Wu T.F.: A semilinear elliptic problem involving nonlinear boundary condition and sign-changing potential. Electron. J. Differ. Equ. 131, 1–15 (2006)
Wu, T.F.: Multiplicity results for a semilinear elliptic equation involving sign-changing weight function. Rocky Mt. J. Math. (2012) (in press)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Rasouli, S.H., Afrouzi, G.A. & Vahidi, J. A variational approach to a quasilinear multiparameter elliptic system involving the p-Laplacian and nonlinear boundary condition. Arab. J. Math. 1, 347–361 (2012). https://doi.org/10.1007/s40065-012-0035-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40065-012-0035-0