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

Abstract EN

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⊆RN 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 by Δpu=div(|∇u|p-2∇u), the functions a and b are sign changing continuous functions in Ω, 1p and ∞ otherwise.

The properties of the first eigenvalue λ1+(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≤λ1+(α), the original problem admits at least one positive solution, while if λ1+(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.