Nonlinear elliptic involving the p-Laplacian problems and stability of positive solutions

where 4sz = div(| ∇z |s−2 ∇z) denotes the p-Laplacian operator; s > 2 and λ is a positive parameter, m > 1 is a constant, Ω ⊂ RN (N ≥ 1) is a bounded region with smooth boundary ∂Ω, weight a(x) satisfies a(x) ∈ C(Ω̄) and we assume that either a(x) > 0 or a(x) < 0 for all x ∈ Ω. Existence of solutions in this type of models (the case system, p = 2) have been studied in [3]. A large number of works have been made studying stability of solutions in the case when m = 1 (see [1, 2, 4, 5]). In [1], authors have shown that every non-negative stationary solution of boundary value problem { −4pu(x) = λg(x)f(u(x)), x ∈ Ω Bu(x) = 0, x ∈ ∂Ω