Global asymptotic stability of bifurcating, positive equilibria of p-Laplacian boundary value problems with p-concave nonlinearities

Abstract

We consider the initial value problem vt=Δp(v)+λg(x,v)ϕp(v),in Ω×(0,∞),v=0,in ∂Ω×(0,∞),v=v0⩾0,in Ω×{0}, where Ω⊂RN, N⩾1, is a bounded domain with smooth boundary ∂Ω ϕp(s):=|s|p−1sgns, s∈R, Δp denotes the p-Laplacian, with p>max⁡{2,N}, v0∈C00(Ω‾), with v0⩾0 on Ω‾ and λ>0. The function g:Ω‾×R→(0,∞) is C0 and, for each x∈Ω‾ the function g(x,⋅):[0,∞)→(0,∞) is Lipschitz and decreasing. With these hypotheses, (IVP) has a unique, positive solution. For each λ>0, (IVP) has the trivial solution v≡0. In addition, there exists 0