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

Abstract

We consider the initial value problem(IVP)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<λmin(g)<λmax(g) (λmax(g) may be ∞) such that:•if λ∉(λmin(g),λmax(g)) then (IVP) has no non-trivial, positive equilibrium;•if λ∈(λmin(g),λmax(g)) then (IVP) has a unique, non-trivial, positive equilibrium eλ∈W01,p(Ω). We prove the following stability results (‘stability’ means with respect to the set of non-trivial, positive solutions):•if 0<λ<λmin(g) then the trivial solution is globally asymptotically stable;•if λmin(g)<λ<λmax(g) then eλ is globally asymptotically stable;•if λmax(g)<λ then any non-trivial, positive solution blows up in finite time.