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

We study the following quasilinear problem with nonlinear boundary condition −Δpu − λa x u|u|p−2 b x u|u|γ−2, in Ω and 1 − α |∇u|p−2 ∂u/∂n αu|u|p−2 0, on ∂Ω, where Ω ⊆ R is a connected bounded domain with smooth boundary ∂Ω, the outward unit normal to which is denoted by n.Δp is the p-Laplcian operator defined byΔpu div |∇u|p−2∇u , the functions a and b are sign changing continuous functions inΩ, 1 < p < γ < p∗, where p∗ Np/ N−p ifN > p and∞ otherwise. The properties of the first eigenvalue λ 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 λ ≤ λ 1 α , the original problem admits at least one positive solution, while if λ 1 α < λ < λ∗, for a positive constant λ∗, 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.