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

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