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Abstract
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In this note, we discuss the stability and instability results of positive solutions for the following reaction-diffusion equation −Bu ∆p= 0 on u = m(x)up−1 − uγ−1 − ch(x) in Ω∂,Ω, (0.1) where ∆ pu = div(|∇u|p−2∇u) is the p-Laplacian operator, p > 1, γ < p, Ω is a bounded domain in RN(N ≥ 1) with smooth boundary, Bu(x) = αg(x)u(x) + (1 − α) ∂u∂n (x) where α ∈ [0, 1] is a constant, g : ∂Ω −→ R+ with g = 1 when α = 1, i.e., the boundary condition may be of Dirichlet, Neumann or mixed type, c is a positive constant and the weight functions m(x) satisfies m(x) ∈ C(Ω) and m(x) > 1 for x ∈ Ω and h : Ω −→ R is a C1,α(Ω) function satisfying h(x) > 0 for x ∈ Ω, max h(x) = 1 for x ∈ Ω and h(x) = 0 for x ∈ ∂Ω. Here u is the population density m(x)up−1 − uγ−1 represents the logistic growth and ch(x) represents the constant yield harvesting rate. we shall establish that every positive solution is linearly unstable. Thai Journal of Mathematic
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