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International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 419341, 11 pages
Research Article

A Multiplicity Result for Quasilinear Problems with Nonlinear Boundary Conditions in Bounded Domains

Department of Mathematics, Faculty of Basic Sciences, Babol Noshirvani University of Technology, 47148-71167 Babol, Iran

Received 28 September 2011; Accepted 2 November 2011

Academic Editor: Enrico Obrecht

Copyright © 2011 S. Khademloo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study the following quasilinear problem with nonlinear boundary condition βˆ’ Ξ” 𝑝 𝑒 βˆ’ πœ† π‘Ž ( π‘₯ ) 𝑒 | 𝑒 | 𝑝 βˆ’ 2 = 𝑏 ( π‘₯ ) 𝑒 | 𝑒 | 𝛾 βˆ’ 2 , in Ξ© and ( 1 βˆ’ 𝛼 ) | βˆ‡ 𝑒 | 𝑝 βˆ’ 2 ( πœ• 𝑒 / πœ• 𝑛 ) + 𝛼 𝑒 | 𝑒 | 𝑝 βˆ’ 2 = 0 , on  πœ• Ξ© , where Ξ© βŠ† 𝑅 𝑁 is a connected bounded domain with smooth boundary πœ• Ξ© , the outward unit normal to which is denoted by 𝑛 . Ξ” 𝑝 is the 𝑝 -Laplcian operator defined by Ξ” 𝑝 𝑒 = d i v ( | βˆ‡ 𝑒 | 𝑝 βˆ’ 2 βˆ‡ 𝑒 ) , the functions π‘Ž and 𝑏 are sign changing continuous functions in Ξ© , 1 < 𝑝 < 𝛾 < 𝑝 βˆ— , where 𝑝 βˆ— = 𝑁 𝑝 / ( 𝑁 βˆ’ 𝑝 ) if 𝑁 > 𝑝 and ∞ otherwise. The properties of the first eigenvalue πœ† + 1 ( 𝛼 ) and the associated eigenvector of the related eigenvalue problem have been studied in (Khademloo, In press). In this paper, it is shown that if πœ† ≀ πœ† + 1 ( 𝛼 ) , the original problem admits at least one positive solution, while if πœ† + 1 ( 𝛼 ) < πœ† < πœ† βˆ— , for a positive constant πœ† βˆ— , it admits at least two distinct positive solutions. Our approach is variational in character and our results extend those of Afrouzi and Khademloo (2007) in two aspects: the main part of our differential equation is the 𝑝 -Laplacian, and the boundary condition in this paper also is nonlinear.