Stability of Positive Weak Solution for Generalized Weighted p-Fisher-Kolmogoroff Nonlinear Stationary-State Problem

In the present paper, we investigate the stability results of positive weak solution for the generalized Fisher–Kolmogoroff nonlinear stationary-state problem involving weighted p-Laplacian operator −d∆P,pu = ka(x)u[ν − υu] in Ω, Bu = 0 on ∂Ω, where ∆P,p with p > 1 and P = P(x) is a weight function, denotes the weighted p-Laplacian defined by ∆P,pu ≡ div[P(x)|∇u|p−2∇u], the continuous function a(x): Ω → R satisfies either a(x) > 0 or a(x) < 0 for all x ∈ Ω, d, k, ν and υ are positive parameters and Ω ⊂ RN is a bounded domain with smooth boundary Bu = δh(x)u + (1 − δ) ∂u/∂n where δ ∈ [0, 1], h: ∂Ω → R+ with h = 1 when δ = 1.