A variational approach to a quasilinear elliptic problem involving the pp-Laplacian and nonlinear boundary condition

Using the technique of Brown and Wu [K.J. Brown, T.F. Wu, A semilinear elliptic system involving nonlinear boundary condition and sign changing weight function, J. Math. Anal. Appl. 337 (2008) 1326–1336], we present a note on the paper [T.F. Wu, A semilinear elliptic problem involving nonlinear boundary condition and sign-changing potential, Electron. J. Differential Equations 131 (2006) 1–15] by Wu. Indeed, we extend the multiplicity results for a class of semilinear problems to the quasilinear elliptic problems of the form: {−Δpu+m(x)|u|p−2u=λa(x)|u|q−2u,x∈Ω,|∇u|p−2∂u∂n=b(x)|u|r−2u,x∈∂Ω. Here ΔpΔp denotes the pp-Laplacian operator defined by Δpz=div(|∇z|p−2∇z), 1pN>p, p∗=∞p∗=∞ if N≤pN≤p), Ω⊂RNΩ⊂RN is a bounded domain with smooth boundary, ∂∂n is the outer normal derivative, λ∈R∖{0}λ∈R∖{0}, the weight m(x)m(x) is a bounded function with ‖m‖∞>0‖m‖∞>0 and a(x),b(x)a(x),b(x) are continuous functions which change sign in Ω¯.