multiple positive solutions of a nonlinear boundary value problem involving a sign-changing weight

in this paper, we study the multiplicity of positive solutions for the following elliptic equation:

{ δu − u = 0 in rⁿ₊,

∂u/∂ν= λa (x) |u|^q−2 u + b (x) |u|^p−2 u on ∂rⁿ₊,

where 1 < q < 2 < p < 2_∗ (2_∗ = 2(n−1)/n−2 if n ≥ 3, 2∗ = ∞ if n = 2),rⁿ₊ = {(x′, x_n ) ∈ rⁿ⁻¹ × r | xn > 0} is an upper half space in rⁿ , λ > 0 and the functions a and b are satisfying some suitable conditions. using the decomposition of the nehari manifold, we prove that the equation has at least two positive solutions provided λ is sufficiently small.