A variational approach to a quasilinear elliptic problem involving the [formula omitted]-Laplacian and nonlinear boundary condition

Abstract

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: { − Δ p u + m ( x ) | u | p − 2 u = λ a ( x ) | u | q − 2 u , x ∈ Ω , | ∇ u | p − 2 ∂ u ∂ n = b ( x ) | u | r − 2 u , x ∈ ∂ Ω . Here Δ p denotes the p -Laplacian operator defined by Δ p z = div ( | ∇ z | p − 2 ∇ z ) , 1 < q < p < r < p ∗ ( p ∗ = p N N − p if N > p , p ∗ = ∞ if N ≤ p ), Ω ⊂ R N is a bounded domain with smooth boundary, ∂ ∂ n is the outer normal derivative, λ ∈ R ∖ { 0 } , the weight m ( x ) is a bounded function with ‖ m ‖ ∞ > 0 and a ( x ) , b ( x ) are continuous functions which change sign in Ω ¯ .