Large solutions for cooperative logistic systems: existence and uniqueness in star-shaped domains

This paper studies the uniqueness of the solution of the singular elliptic problem

{−Δui=∑j=1naij(x)uj−bi(x)fi(ui)in Ω,ui=+∞on ∂Ω,1≤i≤n,$$\begin{array}{} \displaystyle \left\{ \begin{array}{*{20}{l}} { - \Delta {u_i} = \mathop \sum\limits_{j = 1}^n {a_{ij}}(x){u_j} - {b_i}(x){f_i}({u_i})}&{{\rm{in\;}}\,\,{\rm{\Omega }},}\\ {{u_i} = + \infty }&{{\rm{on\;}}\,\,\partial {\rm{\Omega }},} \end{array}\right. \,1 \le i \le n, \end{array}$$(1)

where Ω is a bounded subdomain of ℝ^N, N ≥ 1 whose boundary is sufficiently regular (e.g. of class 𝒞¹) and the heterogeneous terms satisfy aij,bi∈C(Ω¯)$\begin{array}{} \displaystyle {a_{ij}},{b_i} \in {\mathscr C}\left( {\overline \Omega } \right) \end{array}$ for all 1 ≤ i, j ≤ n,

aij(x)>01≤i≠j≤n,bi(x)>01≤i≤n,for allx∈Ω.$$\begin{array}{} \displaystyle \begin{array}{*{20}{l}} {{a_{ij}}(x) > 0}&{1 \le i \ne j \le n,}\\ {{b_i}(x) > 0}&{1 \le i \le n,} \end{array}\quad {\rm{for\;all}}\,\,x \in {\rm{\Omega }}. \end{array}$$(2)

As far as concerns the nonlinear terms, it is assumed that f_i ∈ 𝒞¹[0, ∞) is a nondecreasing function such that f_i(0) = 0, for all 1 ≤ i ≤ n. A function u := (u₁,...,u_n) ∈ [𝒞^2+v(Ω)]ⁿ, v ∈ (0,1), is a solution of (1) if it satisfies the system of (1) and

limx→zx∈Ω,z∈∂Ωui(x)=+∞,1≤i≤n.$$\begin{array}{} \displaystyle \mathop {\lim }\limits_{\begin{array}{*{20}{c}} {x \to z}\\ {x \in {\rm{\Omega }},z \in \partial {\rm{\Omega }}} \end{array}} {u_i}(x) = + \infty ,\quad 1 \le i \le n. \end{array}$$

These functions will be referred as large solutions. Its study goes back to the pioneering works of J. B. Keller [8] and R. Osserman [20], who considered the case of the single equation Δu = f(u) when f isa monotone function. Since them, many works have dealt with large solutions of elliptic equations (see, e.g. the lists of references of [11]), but almost all of them are focused in the study of the single equation. Some of the few existing references for systems are [6,7,12-14], although, except [12], they only study the special case in which n = 2. Moreover, the majority of the works are restricted to the case in which the nonlinearities are of power-type.

In order to ensure the existence of large solutions of (1) one should ask for the following Keller-Osserman type condition:

(KO) There exists f ∈ 𝒞¹ [0, +∞) such that f(u) < f_i(u) for every 1 ≤ i ≤ n and, for every a, b > 0,

I(c):=∫c+∞dθ∫cθbf(s)−asds<+∞for somec>0.$$\begin{array}{} \displaystyle I(c): = \smallint _c^{ + \infty }\frac{{d\theta }}{{\sqrt {\smallint _c^\theta bf(s) - asds} }}\lt { + \infty \quad {\rm{for \; some}}\,\,c} \gt 0. \end{array}$$

This condition is a generalization of the one made by P. Álvarez-Caudevilla and J. López-Gómez in [1]. According to S. Dumont et al. [4], by the monotonicity of f we have that

liminfc→+∞I(c)=0,$$\begin{array}{} \displaystyle \mathop {\lim {\rm{ }}\inf {\rm{ }}}\limits_{c \to + \infty } I(c) = 0, \end{array}$$

so the second part of the Keller-Osserman condition described in [1] is satisfied. The reader is sent to [11, Chapter 3] and [4] for a detailed discussion on Keller-Ossermann conditions. Essentially, (KO) is a condition on the growth of f at infinity. It is not hard to check that the existence of p > 1 and C > 0 such that

fi(u)≥Cup,1≤i≤n,$$\begin{array}{} \displaystyle {f_i}(u) \ge C{u^p},\quad 1 \le i \le n, \end{array}$$(3)

for every u > 0 sufficiently large, entails (KO).

The assumption made on the first inequalities of (2) guarantees that the system of (1) is cooperative, so the maximum principle for cooperative systems of J. López-Gómez and M. Molina-Meyer, [16], which is a fundamental tool for the analysis carried out in this paper, is available. Thus, the usual comparison principle works in our context (see Theorem 2 of Section 2). In particular, we can adapt the construction of a maximal and a minimal large solution given in [1, Sections 3 and 4]. Then, under the general hypotheses of this paper and (KO), there exists a minimal and a maximal solution to (1) for every subdomain Ω ⊂ ℝ^N with ∂Ω sufficiently regular.

The uniqueness of solutions of (1) is still a widely open question, even when (1) reduces to a single equation, and the usual uniqueness argument for the equation via the blow-up rates does not have a trivial extension to cover (1) (see [14]). Our main result establishes the uniqueness of large solution of (1) when Ω is a star-shaped domain, i.e. when there exists a point x₀ ∈ Ω, called the center of Ω, such that the line segment between x₀ and x belongs to Ω for every x ∈ Ω. It can be stated as follows.

Theorem 1 is a substancial extension of [12, Theorem 1.1] and [15, Theorem 1.1]. Indeed, the main result of [12] establishes the uniqueness of solution for the radially symmetric counterpart of (1) with constant coupling coefficients, a_ij ∈ ℝ₊, 1 ≤ i ≠ j ≤ n, while [15, Theorem 1.1] only deals with (1) in the restricted case n = 1 and a₁₁ ∈ ℝ. On the other hand, the case where Ω is an annular region is covered in [12] and [15].

The hypothesis (5) goes back to [10, Eq. (11)] and [3, Eq. (6)]. It is easily seen that (5) implies (KO), which ensures the existence of positive solutions of (1): Assuming (5), we have that (3) is satisfied for the choice p := min{p_i : 1 ≤ i ≤ n} and C := min{f_i(1) : 1 ≤ i < n}. Moreover, (5) implies that

fi(u)/uis increasing,1≤i≤n,$$\begin{array}{} \displaystyle {f_i}(u)/u\;\;{\rm{is increasing,}}\quad 1 \le i \le n, \end{array}$$(6)

because

fi(θu)−fi(u)θu−u≥θpifi(u)−fi(u)θu−u=θpi−1θ−1fi(u)u>fi(u)u,1≤i≤n,$$\begin{array}{} \displaystyle \frac{{{f_i}(\theta u) - {f_i}(u)}}{{\theta u - u}} \ge \frac{{{\theta ^{{p_i}}}{f_i}(u) - {f_i}(u)}}{{\theta u - u}} = \frac{{{\theta ^{{p_i}}} - 1}}{{\theta - 1}}\frac{{{f_i}(u)}}{u} > \frac{{{f_i}(u)}}{u},\quad 1 \le i \le n, \end{array}$$

for every θ > 1 and u > 0. The last inequalities entail that fi′(u)≥fi(u)/u$\begin{array}{} \displaystyle f_i^\prime (u) \ge {f_i}(u)/u \end{array}$, and hence, (f_i(u)/u)′ ≥ 0 for all 1 ≤ i ≤ n. In the proof of Theorem 1, we will not use (5) directly, but the following equivalent condition.

∃α>0such thatρ2+αfi(ρ−αv)≤fi(v)forallρ>1,v>0,1≤i≤n.$$\begin{array}{} \displaystyle \exists \alpha > 0\,\,{\rm{such\;that}}\,\,{\rho ^{2 + \alpha }}{f_i}({\rho ^{ - \alpha }}v) \le {f_i}(v)\quad {\rm{for all}}\,\,\rho > 1,\,\,v > 0,\,\,1 \le i \le n. \end{array}$$(7)

Indeed, the change of variables

v=tu,t=ρα,α=2p−1,$$\begin{array}{} \displaystyle v = tu,\quad t = {\rho ^\alpha },\quad \alpha = \frac{2}{{p - 1}}, \end{array}$$

transforms (5) into (7).

The assumption (4) is a condition on the growth of the heterogeneous terms, a_ij, b_i, along the rays of Ω as x approximates ∂Ω. More precisely, if for every z ∈ ∂Ω we define the functions

aijz(t):=aij(tz),biz(t):=bi(tz),t∈[0,1],1≤i,j≤n,$$\begin{array}{} \displaystyle \begin{array}{*{20}{l}} {a_{ij}^z(t): = {a_{ij}}(tz),}\\ {b_i^z(t): = {b_i}(tz),} \end{array}\quad t \in [0,1],\;\;1 \le i,j \le n, \end{array}$$

then, the second line of (4) means that biz(t)$\begin{array}{} \displaystyle b_i^z(t) \end{array}$ is non-increasing when t ~ 1, for every 1 ≤ i ≤ n. Note that, if the following conditions are satisfied,

aiiz(t)≥0,aijz(t)≤aijz(s)for allt0<t<s<1,z∈∂Ω,1≤i,j≤n,$$\begin{array}{} \displaystyle \begin{array}{l} a^z_{ii}(t)\geq0,\\ a^z_{ij}(t)\leq a^z_{ij}(s) \end{array}\quad \hbox{for all}\;\; t_0<t<s<1,\;\;z\in\partial\Omega,\;\; 1\leq i,j\leq n, \end{array}$$

for some t₀ ∈ (0,1), then the inequalities of the first line of (4) hold. It is remarkable that (4) is weaker than hypothesis (ii) established in [12, Remark 4.2] if we assume Ω is a ball. Indeed, suppose Ω = B_R(0) := {x ∈ ℝ^N : ||x|| < R} and

aij(x):=aij*(dist(x,∂Ω)),1≤i≠j≤n,$$\begin{array}{} \displaystyle {a_{ij}}(x): = a_{ij}^*({\rm{dist}}(x,\partial {\rm{\Omega }})),\quad 1 \le i \ne j \le n, \end{array}$$

for some positive continuous non-increasing functions aij*$\begin{array}{} \displaystyle a_{ij}^* \end{array}$. Then,

ρ2aij(ρx)>aij(ρx)=aij*(dist(ρx,∂Ω))≥aij*(dist(x,∂Ω))=aij(x)$$\begin{array}{} \displaystyle {\rho ^2}{a_{ij}}(\rho x) > {a_{ij}}(\rho x) = a_{ij}^*({\rm{dist}}(\rho x,\partial {\rm{\Omega }})) \ge a_{ij}^*({\rm{dist}}(x,\partial {\rm{\Omega }})) = {a_{ij}}(x) \end{array}$$

for every 1 ≤ i ≠ j ≤ n, ρ > 1 and x ∈ Ω with ρx ∈ Ω.

The distribution of this paper is the following. Section 2 sketches the existence of a minimal and a maximal solution to (1) and provides us with some necessary results for proving the main result. Finally, in Section 3 we show the proof Theorem 1.

The existence of a minimal and a maximal solution to (1) can be obtained simply by adapting the abstract results of J. López-Gómez and P. Álvarez-Caudevilla [1, Section 3] to the case of n equations. For our purpose, it is enough if we show a scheme of this construction, with special attention in the construction of a supersolution for the non singular counterpart of (1).

Given a regular subdomain D ⊂ ℝ^N, we define the operator L:[C2+ν(D)]n→[Cν(D)]n$\begin{array}{} \displaystyle \mathfrak{L}:[\mathscr{C}^{2+\nu}(D)]^n\rightarrow [\mathscr{C}^{\nu}(D)]^n \end{array}$ by

(Lu)i=−Δui−∑j=1naij(⋅)uj,1≤i≤n.$$\begin{array}{} \displaystyle {\mathfrak{L}(u)_i} = - {\rm{\Delta }}{u_i} - \mathop \sum\limits_{j = 1}^n {a_{ij}}( \cdot ){u_j},\quad 1 \le i \le n. \end{array}$$

Thanks to the cooperative structure of L$\begin{array}{} \displaystyle \mathfrak{L} \end{array}$, given by (2), it is well known that there exists a unique σ ∈ ℝ such that the linear eigenvalue problem

{Lφ=σφin D,φ=0on ∂D,$$\begin{array}{} \displaystyle \left\{ \begin{array}{*{20}{l}} {\mathfrak{L}\varphi = \sigma \varphi }&{{\rm{in\;}}\,\,D,}\\ {\varphi = 0}&{{\rm{on\;}}\,\,\partial D,} \end{array}\right. \end{array}$$

admits apositive eigenfunction φ ∈ [𝒞^2+v(D)]ⁿ, φ_i(x) > 0 for all x ∈ D, 1 ≤ i ≤ n. This value σ ∈ ℝ is called the principal eigenvalue of L$\begin{array}{} \displaystyle \mathfrak{L} \end{array}$ under Dirichlet homogeneous boundary conditions. From here on, it will be denoted by σ[L,D]$\begin{array}{} \displaystyle \sigma [\mathfrak{L},D] \end{array}$.

The following theorem goes back to M. Molina-Meyer, [17-19]. It can be obtained by adapting the classical theory of sub and supersolution provided in H. Amann [2] and the maximum principle for cooperative systems of J. López-Gómez and M. Molina-Meyer, [16].

From Theorem 2 we deduce that the mapping

(0,+∞)→[C2+ν(Ω¯)]nm↦θ[Ω,m→],$$\begin{array}{} \displaystyle \begin{array}{*{20}{c}} {(0, + \infty ) \to {{\left[ {{{\mathscr C}^{2 + \nu }}({\rm{\bar \Omega }})} \right]}^n}}\\ {m \mapsto {\theta _{[{\rm{\Omega }},\vec m]}},} \end{array} \end{array}$$

where m→:=(m,…,m)$\begin{array}{} \displaystyle \vec m: = (m, \ldots ,m) \end{array}$, is increasing. Moreover, by adapting the construction provided in [11, Chapter 3] for the single equation, we obtain the following result, which in case n = 2 is given by [1, Theorem 4.10].