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

We study the stability of non-negative stationary solutions of 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 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).