Uniqueness of nonzero positive solutions of Laplacian elliptic equationsarising in combustion theory

Uniqueness of nonzero positive solutions of a Laplacian elliptic equation 
arising in combustion theory 
is of great interest in combustion theory 
since it can be applied to determine where the extinction phenomenon occurs. 
We study the uniqueness 
whenever the orders of the reaction rates are in $(-\infty,1]$. 
Previous results on uniqueness treated the case when the orders 
belong to $[0,1)$. When the orders 
are negative or 1, it is physically meaningful and the bimolecular reaction rate corresponds to the order 1, 
but there is little study on uniqueness. 
Our results on the uniqueness are completely new when the orders 
are negative or 1, and also 
improve some known results when the orders 
belong to $(0,1)$. Our results provide exact intervals of the Frank-Kamenetskii parameters 
on which the extinction phenomenon never occurs. 
The novelty of our methodology is to combine and utilize the results 
from Laplacian elliptic inequalities and equations to derive new results on 
uniqueness of 
nonzero positive solutions for general Laplacian elliptic equations.


(Communicated by Yuan Lou)
Abstract. Uniqueness of nonzero positive solutions of a Laplacian elliptic equation arising in combustion theory is of great interest in combustion theory since it can be applied to determine where the extinction phenomenon occurs. We study the uniqueness whenever the orders of the reaction rates are in (−∞, 1]. Previous results on uniqueness treated the case when the orders belong to [0, 1). When the orders are negative or 1, it is physically meaningful and the bimolecular reaction rate corresponds to the order 1, but there is little study on uniqueness. Our results on the uniqueness are completely new when the orders are negative or 1, and also improve some known results when the orders belong to (0, 1). Our results provide exact intervals of the Frank-Kamenetskii parameters on which the extinction phenomenon never occurs. The novelty of our methodology is to combine and utilize the results from Laplacian elliptic inequalities and equations to derive new results on uniqueness of nonzero positive solutions for general Laplacian elliptic equations. Physically, the reaction rate with m = 0 is called the Arrhenius reaction rate while the bimolecular reaction rate corresponds to m = 1, see [3,7,28,30]. The reaction rate with m < 0 is physically meaningful and has been widely studied, see [21,24,27,31] for m = −1 or m = −2, [27] for m ∈ [−2, 2.67] and [11,21] for m ∈ [−10.31, 2.81].
The study on uniqueness of positive solutions of Eq. (1.1) is of great interest in combustion theory since the extinction phenomenon occurs at the critical value ν E , where ν E = sup ν * > 0 : (1.1) has a unique solution for each ν ∈ (0, ν * ) , see [26]. There has been progress in finding an approximation value smaller than or equal to ν E for suitable n, ε and m. Taira [26] showed that when n ≥ 2 and is a small and computable constant (refer to [26,Theorem 1.5]). The latter case implies that Λ(ε, m) ≤ ν E and Λ(ε, m) is an approximation value to ν E . When n = 1, 2, m ∈ [0, 1) and Ω is the unit open ball in R n , Du [7] proved that We refer to [5,7,8,29,30] for the study on the S-shaped solution curves of Eq. (1.1) when n = 1 or n = 2, m ∈ (−∞, 1) and ε is sufficiently small. The S-shaped solution curves of Eq. (1.1) imply that there exists a number λ 1 (ε) ∈ (0, ∞) such that Eq. (1.1) has a unique nonzero positive solution for ν ∈ 0, λ 1 (ε) . When n ≥ 2 and m ∈ (−∞, 0) ∪ {1}, there is little study on ν E and its approximations. Indeed, the existence of λ 1 (ε) does not provide a good approximation to the critical value ν E .
The novelty of our method is to combine the above result and a known result on existence of nonzero positive solutions for the Laplacian elliptic equation (1.3) obtained in [14] to derive a new result on uniqueness of nonzero positive solutions for Eq. (1.3). In particular, we show that if f is decreasing (may not be strictly decreasing) on R + , Eq. (1.3) has a unique nonzero positive solutions for each ν ∈ (0, ∞), and further that if f is not decreasing on R + , there exists a computable constant ν 1 (ε, m) > 0 such that Eq. (1.3) has a unique nonzero positive solutions for each ν ∈ (0, ν 1 (ε, m)). The

Uniqueness of nonzero positive solutions for Laplace elliptic equations.
In this section, we study uniqueness of nonzero positive solutions for Laplacian elliptic equations of the form , Ω is a bounded open set in R n (n ≥ 1) with a measure µ := meas(Ω) > 0 and with a smooth boundary ∂Ω, ν > 0 is a parameter, and f : R + → R + is continuous.
Uniqueness of positive solutions of Eq. (2.1) was studied under the assumption that the function g defined in (1.4) is either strictly decreasing [2,4,6,10,20,25] or strictly increasing [9,22,23]. However, there are functions f arising in real applications which do not satisfy these conditions. The reaction rate function f in Eq. (1.1) is an example of such functions, see section 3 for detailed study and Figure  1 in [26], where it is showed that the function g with the reaction rate function f in Eq. (1.1) is neither decreasing nor increasing on R + .
In the following we shall prove new results on uniqueness under a different condition imposed on f [see (2.7)], which is weaker than the well-known Lipschitz condition and is satisfied by the reaction rate function f in Eq. (1.1).
We denote by C(Ω) the Banach space of continuous functions defined in Ω. Let be the standard positive cone in C(Ω). By a positive solution to Eq. (2.1), we mean a function u in P which satisfies Eq. (2.1).
We denote by µ 1 the largest characteristic value of the linear Laplacian elliptic equation By [19, (2.18)], µ 1 can be computed by on Ω is the positive cone in the Sobolev space We first state the following result on existence of nonzero positive solutions of Eq.
Lemma 2.1. Assume that f satisfies the following condition:

5)
where f ∞ and f 0 are the same as those defined in (2.4). Then Eq. (2.1) has a nonzero positive solution in P for n ≥ 1 and ν ∈ µ1 f0 , µ1 f ∞ . Let We prove the following result on uniqueness of Laplacian elliptic inequalities.
Lemma 2.2. Assume that f : R + → R + is continuous and there exists a number b ∈ (0, ∞) such that Then if n ≥ 3, then for each ν ∈ 0, ν0 b , where ν 0 is the same as in (2.6), the following Laplacian elliptic inequality has a unique positive weak solution in P 1 , that is, there exists a unique u ∈ P 1 such that where ∇u(x) = ∂u ∂x 1 , · · · , ∂u ∂x n .
and condition (2.7) holds. Moreover, if f ∞ = 0, then Hence, condition (2.10) holds and ν ∈ µ1 f0 , min ν0 b , µ1 Proof. Since m ∈ (−∞, 1] and ε > 0, we have lim u→0 + f (u) u = ∞ and Therefore, the results (1) and (2)  a.e. on Ω, u(x) = 0 on ∂Ω, (3.3) where L is a strongly uniformly elliptic differential operator and the boundary condition involves a first order boundary operator. Note that Theorem 3.1 (1) allows n ≥ 1 and m < 0. Hence, Theorem 3.1 (1) and its generalization to It is known that the extinction phenomenon occurs at the critical value ν E if ν E < ∞ and never occurs if ν E = ∞ (refer to [26]). In the following, we find the regions of (ε, m) on which ν E = ∞. For some (ε, m), it seems difficult to find the exact value ν E . In the latter case, we provide a subinterval (0, ν(ε, m)) of (0, ν E ) and so, the computable value ν(ε, m) is an approximation value to ν E .