Stability properties of non-negative solutions to a non-autonomous p-Laplacian equation

We study the stability of non-negative stationary solutions of-Δpu=λg(x,u),x∈Ω,Bu=0,x∈∂Ω,where Δp denotes the p-Laplacian operator defined by Δpz = div(∣∇z∣p−2∇z); p > 2, Ω is a bounded domain in RN(N ⩾ 1) with smooth boundary Bu(x)=αh(x)u+(1-α)∂u∂n where α∈[0,1],h:∂Ω→R+ with h = 1 when α = 1, λ > 0, and g:Ω×[0,∞)→R is a continuous function. If g(x, u)/up−1 be strictly increasing (decreasing), we provide a simple proof to establish that every non-trivial non-negative solution is unstable (stable).