Multiplicity and uniqueness of positive solutions for elliptic equations with nonlinear boundary conditions arising in a theory of thermal explosion

In this paper we study a model of thermal explosion which is described by positive solutions to the boundary value problem{−Δu=λf(u),x∈Ω,n⋅∇u+c(u)u=0,x∈∂Ω, where f,c:[0,∞)→(0,∞)f,c:[0,∞)→(0,∞) are C1C1 and C1,γC1,γ non decreasing functions satisfying limu→∞f(u)u=0, ΩΩ is a bounded domain in RNRN with smooth boundary ∂Ω∂Ω and λ>0λ>0 is a parameter. Using the method of sub and super-solutions we show that the solution of this problem is unique for large and small values of parameter λλ, whereas for intermediate values of λλ solutions are multiple provided nonlinearity ff satisfies some natural assumptions. An example of such nonlinearity which is most relevant to applications and satisfies all our hypotheses is f(u)=exp[αuα+u] for α≫1α≫1.