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

We study the following quasilinear problem with nonlinear boundary condition − Δ ð ‘ ð ‘¢ − 𠜆 ð ‘Ž ( ð ‘¥ ) ð ‘¢ | ð ‘¢ | ð ‘ âˆ’ 2 = ð ‘ ( ð ‘¥ ) ð ‘¢ | ð ‘¢ | ð ›¾ − 2 , in Ω and ( 1 − ð ›¼ ) | ∇ ð ‘¢ | ð ‘ âˆ’ 2 ( 𠜕 ð ‘¢ / 𠜕 ð ‘› ) + ð ›¼ ð ‘¢ | ð ‘¢ | ð ‘ âˆ’ 2 = 0 , on 𠜕 Ω , where Ω ⊆ ð ‘… ð ‘ is a connected bounded domain with smooth boundary 𠜕 Ω , the outward unit normal to which is denoted by ð ‘› . Δ ð ‘ is the ð ‘ -Laplcian operator defined by Δ ð ‘ ð ‘¢ = d i v ( | ∇ ð ‘¢ | ð ‘ âˆ’ 2 ∇ ð ‘¢ ) , the functions ð ‘Ž and ð ‘ are sign changing continuous functions in Ω , 1 < ð ‘ < ð ›¾ < ð ‘ âˆ— , where ð ‘ âˆ— = ð ‘ ð ‘ / ( ð ‘ âˆ’ ð ‘ ) if ð ‘ > ð ‘ and ∞ otherwise. The properties of the first eigenvalue 𠜆 + 1 ( ð ›¼ ) and the associated eigenvector of the related eigenvalue problem have been studied in (Khademloo, In press). In this paper, it is shown that if 𠜆 ≤ 𠜆 + 1 ( ð ›¼ ) , the original problem admits at least one positive solution, while if 𠜆 + 1 ( ð ›¼ ) < 𠜆 < 𠜆 ∗ , for a positive constant 𠜆 ∗ , 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 ð ‘ -Laplacian, and the boundary condition in this paper also is nonlinear.