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

Abstract In this paper we study a model of thermal explosion which is described by positive solutions to the boundary value problem { − Δ u = λ f ( u ) , x ∈ Ω , n ⋅ ∇ u + c ( u ) u = 0 , x ∈ ∂ Ω , where f , c : [ 0 , ∞ ) → ( 0 , ∞ ) are C 1 and C 1 , γ non decreasing functions satisfying lim u → ∞ f ( u ) u = 0 , Ω is a bounded domain in R N with smooth boundary ∂ Ω and λ > 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 λ , whereas for intermediate values of λ solutions are multiple provided nonlinearity f satisfies some natural assumptions. An example of such nonlinearity which is most relevant to applications and satisfies all our hypotheses is f ( u ) = exp [ α u α + u ] for α ≫ 1 .