Steady state solutions for the Gierer-Meinhardt system in the whole space

We are concerned with the study of positive solutions to the Gierer-Meinhardt system $$ \begin{cases} \displaystyle -\Delta u+\lambda u=\frac{u^p}{v^q}+\rho(x)&\quad\mbox{ in }\mathbb{R}^N\, , N\geq 3,\\[0.1in] \displaystyle -\Delta v+\mu v=\frac{u^m}{v^s}&\quad\mbox{ in }\mathbb{R}^N,\\[0.1in] \end{cases} $$ which satisfy $u(x), v(x)\to 0$ as $|x|\to \infty$. In the above system $p,q,m,s>0$, $\lambda, \mu\geq 0$ and $\rho\in C(\mathbb{R}^N)$, $\rho\geq 0$. It is a known fact that posed in a smooth and bounded domain of $\mathbb{R}^N$, the above system subject to homogeneous Neumann boundary conditions has positive solutions if $p>1$ and $\sigma=\frac{mq}{(p-1)(s+1)}>1$. In the present work we emphasize a different phenomenon: we see that for $\lambda, \mu>0$ large, positive solutions with exponential decay exist if $0<\sigma\leq 1$. Further, for $\lambda=\mu=0$ we derive various existence and nonexistence results and underline the role of the critical exponents $p=\frac{N}{N-2}$ and $p=\frac{N+2}{N-2}$.


Introduction and the main results
In 1972, Gierer and Meinhardt [15] derived a mathematical model to account for morphogenesis in hydra, a tissues regeneration process through controlling the spatial distribution of cells.The morphogenesis was first observed by Trembley [29] in 1744 and is directed by two types of factors -chemical molecules and mechanical forces.Based on the findings of Turing [30] in 1952, the Gierer and Meinhardt model assumes the existence of two chemical substances (morphogens): a slowly diffusing activator and a fast diffusing inhibitor.Their original model in [15] reads as (GM) in Ω × (0, T ), in Ω × (0, T ), where Ω ⊂ R N is a smooth and bounded domain, ν denotes the outer unit normal vector at ∂Ω, λ, µ ≥ 0 and ρ ∈ C(Ω) is a nonnegative function.The nonnegative solution (u, v) of the above system defines the concentrations of the activator and inhibitor respectively while ρ is related to the basic production rate of the activator.Finally, the exponents p, q, m, s are assumed to satisfy p > 1 , q > 0 , m > 0 and s > −1.
The inhibitor diffuses much faster compared to the activator, which means that the diffusion coefficients d and D satisfy D ≫ d > 0. As explained in [18], the dynamics of the Gierer-Meinhardt system is characterised by two numbers: • the net self-activation index (p − 1)/m which is responsible for the strength of self-activation of the activator; • the net cross-inhibition index q/(s + 1) which measures how strongly the inhibitor suppresses the production of the activator and that of itself.
From biological interpretations of the above exponents, it is assumed that Under the assumption σ > 1 it was obtained in [16] the existence of global in time solutions to (GM); see also [6,8,17,20,23,26].As already emphasised in [26], various existence and nonexistence results on the system (GM) can be derived according to the case where the basic production rate ρ of the activator is identically zero or not.One particular perspective in understanding the structure of (GM) is given by the study of the shadow system, an approach introduced by Keener [19].More precisely, letting D → ∞ in (GM) we deduce v(x, t) = w(t), t ∈ (0, T ) and so we are lead to consider: in Ω × (0, T ), in Ω × (0, T ), ∂u ∂ν = 0 on ∂Ω × (0, T ), u(x, 0) = u 0 (x), w(0) = w 0 in Ω.
In the present work we investigate the steady state solutions of the Gierer-Meinhardt system (GM) in the whole space.Precisely, we consider the system (1.1) where ρ ∈ C(R N ), ρ ≥ 0, p, q, m, s > 0 and λ, µ ≥ 0. We are looking for positive classical solutions of (1.1), that is, functions u, v ∈ C 2 (R N ) such that u, v > 0 and satisfy (1.1) at every point of R N .In a smooth and bounded domain, the above system was discussed in [11,13,17,21,28,31] under various boundary conditions.In [22] the following related system in R 3 was investigated: (1.2) It is obtained in [22] that for ε > 0 sufficiently small, the system (1.2) admits a positive radial solution (u, v) such that u and v decay exponentially at infinity.Similar results in dimension N = 2 were previously obtained in [9] and [27].
Turning to our system (1.1), we discuss the existence and nonexistence of solutions when ρ ≥ 0. We shall see that only the case p > 1 is relevant to our study as for 0 < p ≤ 1 no positive solutions to (1.1) exist (see Theorem 1.1 below).Furthermore, the existence of solutions which decay exponentially at infinity is closely related to the conditions p > 1 and σ ≤ 1 as for (1.2).
Throughout this paper, for two positive functions f, g ∈ C(R N ) we use the symbol f (x) ≃ g(x) to denote that the quotient f /g is bounded between two positive constants in a neighbourhood of infinity.Also, we say that f ∈ C(R N ) decays exponentially at infinity if f (x) ≃ e −a|x| for some a > 0.
Our first result on (1.1) concerns the case λ, µ > 0 and reads as follows.
(iii) Assume p > 1 ≥ σ, ρ > 0 and ρ(x) ≃ e −a|x| for some a > 0.Then, there exist two positive constants C 1 , C 2 > 0 depending on p, q, m, s and ρ such that the system (1.1) has positive solutions (u, v) decaying exponentially at infinity provided that The constants C 1 and C 2 are described explicitly in the proof of Theorem 1.1 which we present in Section 3. Observe that since σ ≤ 1 one has 3) is positive.In particular, (1.3) states that solutions of (1.1) with exponential decay exist if λ, µ are large enough.
In the following we take λ = µ = 0 in (1.1) and derive a different set of result.We shall thus investigate the system (1.4) We first state the following nonexistence result.
Theorem 1.2.The systems (1.4) has no positive solutions in any of the following cases: Part (iv) in Theorem 1.2 above is a consequence of Theorem 4.1 in Gidas and Spruck [14].The result below provides sufficient conditions for which (1.5) holds.
N −2 and ρ ≡ 0.Then, the system (1.4) has no positive solutions (u, v) which satisfy one of the conditions below: (i) There exists 0 < a < (N −2)s+N m such that u(x) ≃ |x| −a .
(ii) u and v are radial.
(ii) Conversely, for all p > N N −2 there exist q, m, s > 0 and a continuous function ρ ∈ C(R N ), ρ ≥ 0, such that (1.4) has a positive solution.
Similarly, the result below states the optimality of the exponent N +2 N −2 when it comes to the study of radial solutions to (1.4).
N −2 then the system (1.4) has no positive radial solutions for all q, m, s > 0; (ii) Conversely, for all p > N +2 N −2 there exist q, m, s > 0 such that (1.4) has a positive radial solution.
The main tool in deriving the nonexistence of positive solutions to (1.1) and (1.4) is the use of various integral representations of solutions to semilinear elliptic equations in R N which were recently obtained in [7]; we recall the relevant findings of [7] in Section 2.1 below.The existence of positive solutions to (1.1) and (1.4) combines the Schauder fixed point theorem with the study of the single equation where ψ ∈ C(R N ) has a specific decay rate at infinity and µ ≥ 0.
The remaining part of the manuscript is organised as follows.In Section 2 we derive some preliminary results related to integral representations for semilinear elliptic equations in R N .Sections 3 is devoted to the proof of Theorem 1.1.In Section 4 we prove Theorem 1.2 and Corollary 1.3.Finally, the proof of Theorem 1.4 and Corollaries 1.5, 1.6 and 1.7 are given in Section 5.

Fundamental solutions and integral representations
Let s ∈ R and z ∈ C \ {w ∈ C : Re w ≤ 0, Im w = 0}.The modified Bessel functions I s (z) and K s (z) of the first and second kind respectively are defined as (2.1) where Γ(z) is the standard Gamma function.For an integer n, the above definition of K n (z) is understood in the sense K n (z) = lim s→n K s (z).Given λ > 0, the function is the fundamental solution of the operator −∆ + λI (see [4,Section 3]) in the sense that The asymptotic behaviour of G λ is summarised below.(i) There exists c 1 > 1 such that for all |x| > 1.
(ii) There exists c 2 > 1 such that (iv) There exists C > 0 such that for all |x| > 1.
The following result, recently obtained in [7], establishes the analogy between the distributional solutions of (2.5) and their integral representation with kernel G λ .
where G λ is the fundamental solution of −∆ + λI as given in (2.2).Conversely, if then the function u given by (2.7) is a distributional solution of (2.5).
As a consequence, we have: Lemma 2.3.Let λ, µ > 0 and (u, v) be a solution of (1.1).Then, for all x ∈ R N one has To end this section, let us mention that similar results hold for the Laplace operator −∆.Precisely, let (2.10) Then, G 0 is the fundamental solution of −∆ in the sense that The result below was obtained in [5] and extended in [7].
R N → R be a measurable function and ℓ ∈ R. The following statements are equivalent: and for a.a.x ∈ R N , u satisfies (iii) We have and the following representation holds As a consequence of Proposition 2.4 and similar to Lemma 2.3 we have: Lemma 2.5.Let (u, v) be a solution of (1.4).Then, there exists a constant c(N) > 0 such that for all x ∈ R N one has

Some results on semilinear elliptic equations
In this section we collect some useful results on semilinear elliptic equations.
Then, the inequality −∆u ≥ f (u) has no positive C 2 -solutions in any exterior domain of R N , N ≥ 3.
Lemma 2.9.Let s > 0 and ψ ∈ C(R N ) be a positive function such that ψ(x) ≃ e −γ|x| for some γ > 0.Then, for any µ > γ s+1 2 the problem (2.17) Define now Using the estimates (2.16) together with (2.18) we find Thus (v, v) is an ordered pair of sub and super-solution of (2.17).By the standard sub and super-solution method (see, e.g., [12]) for any n ≥ 1 there exists We claim that v n ≤ v n+1 in B n .The inequality clearly holds on ∂B n .Assuming that the set {x ∈ B n : contradiction.Thus, {v n (x)} n≥1 is an increasing sequence bounded from above and from below by v(x) and v(x).We can define v(x) = lim n→∞ v n (x).By standard elliptic arguments we further deduce that v is a solution of (2.17 . The uniqueness of v as a solution of (2.17) follows in the same way as above.
Proof.By direct calculation we have Combining the above estimates we deduce (2.20).
We next establish a counterpart result to Lemma 2.9 for the problem (2.21) Lemma 2.11.Let s > 0 and φ ∈ C(R N ) be a positive function.
(ii) From our assumptions, there exist M > m > 0 such that Take now Since 2 < γ < (N − 2)s + N, we have a ∈ (0, N − 2) and C, c > 0. By taking M > 1 large and m ∈ (0, 1) small, we may assume C > c > 0. By Lemma 2.10 one can check that v(x) = cZ a and v(x) = CZ a satisfy We next proceed as in Lemma 2.9.First, for any n ≥ 1, by the sub and super-solution method there exists Then, v n ≤ v n+1 in B n .Finally, using standard elliptic arguments one has that v(x) = lim n→∞ v n (x) is a solution of (2.21).The uniqueness follows as in the proof of Lemma 2.9.
3 Proof of Theorem 1.1 (i) Assume 0 < p ≤ 1 and there exists (u, v) a positive solution of the system (1.1).Since u(x) → 0 as |x| → ∞, we have for R > 0 large enough that homogeneous Dirichlet boundary condition on ∂Ω R .Denote also by ϕ 1 ∈ C 2 (Ω R ) the corresponding eigenfunction which we normalize as ϕ 1 > 0 in Ω R , that is Then, multiplying by ϕ 1 in the first equation of (1.1) and integrating over Ω R we find Using the Green's formula and the Hopf boundary point lemma we have where ν denotes the outward unit normal vector to ∂Ω R .From (3.1) and (3.2) we find This yields a contradiction for R > 0 large since 2 λ and there exists (u, v) a positive solution of (1.1) with u(x) ≃ e −a|x| for some a > 0. Note that the fundamental solution G λ of the operator −∆ + λI satisfies (2.3).By the maximum principle one has u ≥ cG λ in R N \ B 1 , for some c > 0. Using Lemma 2.1(i) it follows that for all |x| > 1.
Take now λ, µ > 0 large such that For M 1 > M 1 > 0 and M 2 > M 2 > 0 define the set Let us first note that for (u, v) ∈ Σ and (3.5) we have (3.9) This shows that f = u p v q + ρ(x) is bounded and thus (2.8) holds.By Proposition 2.2 we have and this justifies the existence and uniqueness of the solution T u in (3.8).Also, from Lemma 2.9 with ψ(x) = u m (x) ≃ e −γ|x| , γ = am and µ > ( am s+1 ) 2 (see (3.6)) we deduce the existence and the uniqueness of and observe that all fixed points of Ψ are solutions of (1.1).We shall establish the existence of such fixed points (and thus, the existence of a solution to (1.1)) through Schauder fixed point theorem.
Proof.From the estimate (3.9) we have Thus, by Lemma 2.8 and (3.6), the function By the maximum principle and (3.10a) we find Thus, by Lemma 2.8 and the above estimate the function By the maximum principle and (3.10b) one has We now focus on the inequality We first note that ξ = M 2 W b satisfies by (3.10c), (3.6) and the estimate (2.16), the inequality We have thus obtained With the same method as in the proof of the uniqueness of the solution in Lemma 2.9 we deduce As above, from (3.11) and (3.12) we deduce T v ≥ ξ = M 2 W b in R N and this concludes our proof.Proof.We shall prove that if (1.3) holds for some suitable constants C 1 , C 2 > 0, then there are M 1 > M 1 > 0 and M 2 > M 2 > 0 that satisfy slightly more general conditions than (3.10a)-(3.10d),namely Adding the two conditions in (3.13a) we deduce (3.10a).Let us note that from (3.13b) we have Further, from (3.13a) 1 we find and then from (3.13c) one obtains Let us note that by (3.15), condition (3.13a) 2 holds if where c 0 > 0 depends on p, q, m, s and α, β.The same condition but with different constants instead of c 0 arises if we impose M 1 > M 1 and then M 2 > M 2 .This finishes the proof of our lemma.
Proof of Theorem 1.1 completed.Assume that (3.13a)-(3.13d)hold.By the above two lemmas, this implies that Σ is invariant to Ψ. Due to lack of compactness, we cannot apply the Schauder fixed point theorem directly on C(R N ) × C(R N ).In turn, we shall use this result on C(B n )×C(B n ) and combine it with elliptic regularity to deduce the existence of a positive solution to (1.1) in the whole space.Let where M 1 > M 1 > 0 and M 2 > M 2 > 0 are the constants that satisfy (3.13a)-(3.13d).
For (u, v) ∈ Σ n , let ( u, v) ∈ Σ be a fixed extension of (u, v), where Σ is defined in (3.7).Let (U, V ) = (T u, T v) where T u and T v are given by (3.8) (with u, v replaced by u, v).We define Note that by (3.8) we have that (U, V ) satisfies In particular, one has that the above definition of Ψ n is independent on the extension We can thus apply the Schauder fixed point theorem for Ψ n and there exists (u n , v n ) ∈ Σ n a fixed point of Ψ n , which means (u n , v n ) is a solution of the system (3.17) Since b = am/(s + 1) and σ ≤ 1 we have By standard elliptic estimates, it follows that ) for all r > 1, and thus, it is bounded in C 1,γ (B 1 ) × C 1,γ (B 1 ) for some γ ∈ (0, 1).Since the embedding C 1,γ (B 1 ) ֒→ C 1 (B 1 ) is compact, it follows that ).Now, we proceed similarly with the sequence {(u 1 n , v 1 n )} n≥2 which, by elliptic estimates it is bounded in W 2,r (B 2 ) × W 2,r (B 2 ) for all r > 1 and thus, has a convergent subsequence ).This concludes our proof.

Proof of Theorem 1.2
Assume (u, v) is a positive solution of (1.4).
(i) Since v is positive and v(x) → 0 as |x| → ∞, it follows that v −q ≥ C > 0 in R N for some constant C > 0. Thus, u satisfies −∆u ≥ Cu p in R N .By Lemma 2.6 it follows that (1.4) has no solutions for all 0 < p ≤ N/(N − 2).
(ii) From Lemma 2.5 we deduce (iii) By (2.15b) in Lemma 2.5 and (4.1) we find In particular, for x = 0 we find (iv) Observe that u ∈ C 2 (R N ) is positive and satisfies −∆u = h(x)u p in R N , where h = v −q .In light of the second equation of (1.4) one has ∆h = q(q + 1)v −q−2 |∇v| 2 + qv −q−1 (−∆v) ≥ 0 in R N .Also, since v(x) → 0 as |x| → ∞, one has h ≥ c > 0 in R N .Further, thanks to the assumption (iv) in Theorem 1.2 one has |∇h(x)| ≤ C |x| h(x) for |x| > 1 large.By Lemma 2.7 it now follows that u ≡ 0, contradiction.Thus, there exist We shall now check that v satisfies (1.5) in R N \ B 2 which further implies that (1.4) has no positive solutions.By (2.15b) in Lemma 2.5 we deduce From (4.5)-(4.8)we deduce that v satisfies (1.5) and so the are no positive solutions of the system (1.4).
Then, from (4.11) we find ds for all r > 0.
5 Proof of Theorem 1.4 (i) By (1.6) we have ρ(x) ≥ c|x| −a for all x ∈ R N \ B 1 .Thus, if a ≤ 2 it follows that and then by Theorem 1.2(ii) we deduce that the system (1.4) has no solutions.Assume now that 2 < a ≤ 2 1 + 1 m holds.We will check that the integral condition in Theorem 1.2(iii) holds.Let Note that condition (1.8) is equivalent to The existence and uniqueness of T v ∈ C 2 (R N ) follows from Lemma 2.11(ii) with φ(x) = u m (x) ≃ |x| −m(a−2) , γ = m(a − 2) and by (1.7) one has 2 < γ = m(a − 2) < (N − 2)s + N.
To prove the existence of T u we first note that from (1.6), (5.1), (5.2) and the definition of the set Σ one has (5.4) Since a > 2, the function f = u p v q +ρ(x) satisfies (2.13).Indeed, by (5.4) for all x ∈ R N we have

By Proposition 2.4 it follows that
u p (y) v q (y) + ρ(y) dy, and this shows the existence and uniqueness of T u ∈ C 2 (R N ).Define Φ : Σ → C(R N ) × C(R N ) by Φ(u, v) = (T u, T v).Then, fixed points of Φ are solutions of (1.4). .Also, the third inequality in (5.9) is equivalent to Then, if (1.9) is fulfilled, then (5.10)-(5.12)hold, which imply that (5.6a)-(5.6d)and (5.9) hold.By Lemma 5.1 it now follows that Σ is an invariant set of Φ and this concludes the proof.
To finish the proof of Theorem 1.4(ii) we follow the same approach as in the proof of Theorem 1.1(iii).We use the Schauder fixed point theorem on C(B n ) × C(B n ) and then, a diagonal process allows us to deduce the existence of a positive solution to (1.4).
(ii) Assume p > N N −2 , so that 2p p−1 < N. We show that one can find q, m, s and a that satisfy 2 1 + 1 m < a < N, 0 < σ < 1, (1.7) and (1.8).Let a > 0 be close to N such that 2 < 2p p−1 < a < N. By letting m > 0 large enough, one has 2 1 + 1 m < a < N. At this point a is fixed.Now, by taking s > m large we have that (1.7) is fulfilled.Next, by letting q > 0 small one has σ ∈ (0, 1) is small and fulfils (1.8) in Theorem 1.4.It remains to choose β > α > 0 in (1.6) and (1.9) in order to apply Theorem 1.4(ii) and thus to deduce the existence of a positive solution to (1.4).Then u = v = w satisfy (1.4).