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. This paper is concerned with the existence and stability of nontrivial positive steady states of Shigesada-Kawasaki-Teramoto competition model with cross diﬀusion under zero Neumann boundary condition. By applying the special perturbation argument based on the Lyapunov-Schmidt reduction method, we obtain the existence and the detailed asymptotic behavior of two branches of nontrivial large positive steady states for the speciﬁc shadow sys- tem when the random diﬀusion rate of one species is near some critical value. Further by applying the detailed spectral analysis with the special perturba- tion argument, we prove the spectral instability of the two local branches of nontrivial positive steady states for the limiting system. Finally, we prove the existence and instability of the two branches of nontrivial positive steady states for the original SKT cross-diﬀusion system when both the cross diﬀusion rate and random diﬀusion rate of one species are large enough, while the random diﬀusion rate of another species is near some critical value.

1. Introduction and statement of main results. In this paper we investigate the following quasilinear reaction diffusion model with cross-diffusion, which was first proposed by Shigesada-Kawasaki-Teramoto [16] for describing the segregation of two competing species under the intra-and inter-specific population pressure, x ∈ Ω. (1.1) Here u(x, t) and v(x, t) represent the densities of two competing species at the location x and time t, Ω is a bounded region in R N with the smooth boundary ∂Ω and ν is the outward unit normal vector on ∂Ω. We assume d i , a i , b i and c i are positive constants throughout this paper. The coefficients ρ 11 and ρ 22 are the self-diffusion rates which represent the intra-specific population pressures, ρ 12 and ρ 21 denote the cross-diffusion coefficients which measure the population pressure from the competing species. In the following if ρ 12 or ρ 21 is positive, the model (1.1) will be called the SKT competition model with cross diffusion. For ease of notations, we denote A = a 1 a 2 , B = b 1 b 2 and C = c 1 c 2 .
If ρ ij = 0, i, j = 1, 2, (without self-diffusion and cross-diffusion), the competition model (1.1) becomes the typical Lotka-Volterra competition model with random diffusion under zero Neumann boundary condition: It is well known that for any fixed d 1 , d 2 > 0 and any given nonnegative initial data, there exists a unique uniformly bounded global solution to system (1.2) and the asymptotic behaviour of the solution is nearly the same (except the case B < A < C) as that for Lotka-Voltera ODE competition model ( tending to some constant steady state eventually). While for the case B < A < C, it follows from [4] that if Ω is convex then system (1.2) has no stable nontrivial positive steady state.
The SKT competition model with cross diffusion (if ρ 12 = 0 or ρ 21 = 0) is a strongly coupled quasi-linear parabolic system, in the past three decades the SKT competition model with cross diffusion and some biological models with the SKT type of cross diffusion have attracted tremendous attention of both mathematicians and ecologists. Especially for the simplified SKT competition model (1.1) with ρ 11 = ρ 22 = ρ 21 = 0, i.e.
In this paper we shall be more interested in existence and stability analysis of some nontrivial positive steady states for the simplified SKT competition system (1.3). Before stating our work we shall first give a brief survey on some known work on the existence and stability/instability of positive nontrivial steady states for the SKT model (1.1) or (1.3) especially when ρ 12 /d 1 is large enough.
The first important theoretical work on the existence of nontrivial steady states for SKT model is due to [13], in which it was shown that in some strong competition case B < C when d 1 and ρ12 d1 are large enough but d 2 is small, (1.3) admits several types of positive steady states with interior or boundary transition layers; the stability/instability of such steady states was investigated in [3] by applying SLEP method.
The existence/non-existence and the priori estimates of nontrivial positive steady states for SKT model (1.1) with cross-diffusion in multidimensional case were widely and deeply investigated in [8] and [9]. In [9] the authors also proved the related uniform boundedness of nontrivial positive steady states and proposed three types of limiting stationary problems of (1.1) to classify all the possible asymptotic behavior of positive steady states as one of cross-diffusion, say ρ 12 , tends to infinity (See Theorem 1.4 and 4.1 in [9]). By investigating the existence of steady states for some limiting systems in one dimensional case, Lou and Ni [9] also did some pioneering work on the existence of several types of spiky steady states for some related limiting systems and the original SKT competition system (1.3) when ρ 12 /d 1 is large enough but d 2 is small.
For convenience of our later use, we shall restate the related results in [9] (See Theorem 1.4 and Theorem 4.1 in [9]) about all the possible limiting systems of the simplified SKT model (1.3) as ρ 12 → ∞ as follows. Theorem 1. [9] Suppose that N ≤ 3, A = B, A = C and a2 d2 is not equal to any eigenvalue of −∆ with homogeneous Neumann boundary condition on ∂Ω. Let (u i , v i ) be a sequence of positive nontrivial stationary solutions of (1.3) with (d 1 , ρ 12 ) = (d 1,i , ρ 12,i ) and the fixed d 2 > 0, then the following conclusions hold.
(2) If ρ 12, i /d 1, i → ∞ and d 1, i → ∞ as i → +∞, then by passing to a subsequence if necessary, the sequence (1.6) Since the work of Lou and Ni [9], the first limiting system (1.4) (a shadow system of (1.3)) has been deeply and widely investigated by some mathematicians. For the one dimensional case with Ω = (0, 1), Lou, Ni and Yotsutani [11] obtained some nearly optimal results on the existence and non-existence of positive steady states for the first limiting system (1.4), which can be briefly summarized as follows (see Figure 1): If d 2 ≥ a2 π 2 , then the limiting system (1.4) does not have any non-constant solutions; while for 0 < d 2 < a2 π 2 , if A < B < C, or if C < B and A < B+3C

4
, then the limiting system (1.4) has no non-trivial positive steady states; see also the region filled with slashes in Figure 1 in which the existence of positive monotone steady states are proved.  For the case 1 in Figure 1, it was shown in [9] that there exist positive steady states with boundary spike layer near the positive constant steady state (u * , v * ) when d 2 is small enough (see Figure 2(a)). The instability of such spiky steady states was proved in [18].
For the case 2 in Figure 1, the existence and the spiky structure of another type of positive steady states with boundary spike layers (ζ d2 , ψ d2 (x)) for the limiting system (1.4) were investigated in [11] when d 2 is small enough. In [19] for any A > B+3C 4 by applying different approach the authors also proved the existence and the detailed structure of the spiky steady states (ζ d2 , ψ d2 (x)) for the limiting system (1.4) and existence of the corresponding large spiky steady states (u(x), v(x)) perturbed from (ζ d2 /ψ d2 (x), ψ d2 (x)) for the original SKT competition model with large ρ 12 and small d 2 (see Figure 2(b)).
For the case 3 in Figure 1, in [11] it was proved that for any A > B the limiting system (1.4) has a type of nontrivial positive steady states (ξ d2 , ψ d2 (x)) with singular bifurcation structure when d 2 is near a 2 /π 2 , where (ξ d2 , ψ d2 (x)) → (0, 0) and as d 2 → a 2 /π 2 . The existence and the stability of the steady states perturbed from (ξ d2 /ψ d2 (x), ψ d2 (x)) for the original SKT competition system were proved in [15] (see Figure 2(c)). Similar existence and stability results for the shadow system (1.4) and for the original SKT model in multidimensional case were proved in [12].
As far as we know, there are only a few works on the limiting systems (1.5) and (1.6), where the v component of the steady state to the original SKT system (1.1) is assumed to tend to zero but ρ 12 v(x) is positive and bounded as ρ 12 tends to infinity. The first theoretical work based on the investigation of the third limiting system (1.6) is due to Lou and Ni [9], in which for any A > B and small enough d 2 > 0 the authors proved the existence of positive spiky steady states for the limiting system (1.6) and the existence of the corresponding spiky steady states near (a 1 /b 1 , 0) for the SKT model when both d 1 and ρ 12 /d 1 are large enough but d 2 is small enough. Such spiky steady states to the limiting system and the original SKT model were proved to be unstable in [17].
Focusing on the second limiting system (1.5) with the fixed d 1 , recently K. Kuto [5] proved the existence of a local branch of positive steady states (u(x, a 2 ), w(x, a 2 )) with bifurcating structure in multi-dimensional case when a 2 is near a 1 b 2 /b 1 ; and proved the existence of a global branch of positive steady state (u(x, a 2 ), w(x, a 2 )) in one dimensional case with Ω = (0, 1) for any a 2 ∈ (d 2 π 2 , a 1 b 2 /b 1 ), and the global branch of steady states in one dimensional case is proved to having special blowing up structure as a 2 → d 2 π 2 , precisely speaking w(x, a 2 ) → +∞ but u(x, a 2 ) → a2 b2 · as a 2 → d 2 π 2 . Recently the local bifurcating steady states to the limiting system (1.5) were proved to be unstable in [7], in which it is also proved that there exists a local branch of the corresponding unstable positive steady states to the original SKT model (1.1) when ρ 12 is large enough.
In [5] it is remarked that in one dimensional case with Ω = (0, 1) as d 2 → a2 π 2 the w component of the nontrivial positive solution (u(x), w(x)) to the limiting system (1.5) tends to infinity; while the u component tends to a bounded positive function, which is the same limit of u component of the another branch of steady states perturbed from the first limiting system (1.4) as d 2 → a2 π 2 . Motivated by the work of [5], [12] and [15] on the steady states of the limiting systems (1.4) and (1.5) in one dimensional or multidimensional cases when d 2 is near a 2 /λ 1 (λ 1 is the second eigenvalue of −∆ under the homogeneous Neumann boundary condition), in this paper we focus on existence and stability analysis of nontrivial positive steady states to the third limiting system (1.6) and the SKT competition system (1.3) when both d 1 and ρ 12 /d 1 are large enough but d 2 is near a 2 /λ 1 . It is natural to guess that there may also exist some local branches of steady states (τ d2 , w d2 (x)) for the third limiting system when d 2 is near a 2 /λ 1 and which may have similar or different blowing up structure (or bifurcating from infinity) from the recent work in [5] on the second limiting system (1.5), and in this paper we shall apply different approach to investigate the existence and the stability as well as the detailed asymptotic structure of two local branches of blowing up steady states to the limiting system (1.6) and to the original SKT model (1.3) when d 2 is near a 2 /λ 1 .
Before stating our main results, we shall first give a brief derivation of the third limiting system (1.6) and the corresponding evolutional system, which is the limiting system of (1.3) when the v component tends to zero but the limit of ρ12 d1 v is positive and bounded as both ρ12 d1 and d 1 approach the infinity. Denote α = ρ12 d1 and set w = αv, φ = (1 + αv)u, then system (1.3) can be rewritten as (1.7) Introducing s = 1 d1 and r = 1 α , then (1.7) becomes the following system (1.8) Let r → 0 + , then system (1.8) can be reduced to the following system Further we consider the limiting problem of (1.9) as s → 0 + . If we assume that all the quantities in the first equation of (1.9) remain bounded as ∈ Ω due to the boundary condition. It is evident that (τ (t), w(x, t)) satisfies the following limiting system (1.10) Let 0 = λ 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λ i ≤ · · · be the eigenvalues of the following eigenvalue problem and let ϕ 0 (x), ϕ 1 (x), · · ·, ϕ i (x), · · · be the corresponding normalized eigenfunction with min In one dimensional case with Ω = (0, 1), it is well known that all the eigenvalues of (1.11) are simple, and λ 1 = π 2 with a sign changing eigenfunction ϕ 1 (x) = cos(πx). However in multidimensional case for j ≥ 1 λ j may be not simple. In the following of this paper, for the simplicity of proof we always assume that the second eigenvalue λ 1 of (1.11) is simple, thus λ 0 = 0 < λ 1 < λ 2 . In the following, we assume that For multi-dimensional case the existence of the µ ± and the properties of the functions (1 + µ ± ϕ 1 (x)) and g(µ) will be stated in Lemma 2.3 (see also Corollary 1 in [12]). The values µ + and µ − enable us to precisely estimate the limiting profile of the ratio τ w(x) of the steady states to the limiting system (1.10) as d 2 → a 2 /λ 1 , which with the non-degeneracy of g(µ) at µ = µ ± are also useful in proving the existence and the instability of the perturbed nontrivial positive steady states of (1.10) when d 2 is near a2 λ1 . Now we state our main results on the existence and instability of nontrivial positive steady states to the limiting system (1.10).
Theorem 1.1. Assume A > B, N ≤ 4 and λ 1 is simple, then there exists δ 0 > 0 such that for each fixed d 2 ∈ (a 2 /λ 1 − δ 0 , a 2 /λ 1 ), system (1.10) has two nontrivial positive solutions (τ ± d2 , w ± d2 (x)) satisfying By virtue of the detailed asymptotic structure and the spectral results obtained for the limiting system (1.10), by applying perturbation argument we can further prove the existence and the instability of the perturbed steady states to the original SKT model (1.3) when both d 1 and ρ 12 /d 1 are large enough.
Our results for the original SKT model (1.3) are stated as follows.
d1 ≥d, the SKT cross-diffusion system (1.3) has two types of nontrivial positive steady states The present paper is organized as follows. In Section 2, we investigate the existence and the stability of two local branches of blowing up steady states to the limiting system (1.6) and give the detailed proofs of Theorems 1.1 and 1.2. The existence and instability of the nontrivial positive steady states to the original crossdiffusion system (1.3) will be investigated in Section 3.

2.
Existence and instability of nontrivial positive steady states to the limiting system (1.6).
2.1. Existence of nontrivial positive steady states to the limiting system (1.6). In this subsection, we first state some preliminary results on the non-existence of nontrivial positive steady states to the limiting system (1.10).
Lemma 2.1. If A < B, then the limiting system (1.10) has no positive steady states.
Lemma 2.1 can be proved by nearly the same argument as in [5] and Lemma 2.2 can be similarly proved as in [11], here we omit the details of the proofs. Lemmas 2.1 and 2.2 imply that to get the existence of nontrivial positive solutions to the limiting system (1.10), it is necessary to consider the case when 0 < d 2 < a 2 /λ 1 and A > B. In the following of this subsection we shall investigate the existence and the detailed structure of some nontrivial positive steady states to the limiting system (1.10) when a 2 − d 2 λ 1 > 0 is small enough. Let (τ, w(x)) be a nontrivial positive steady state of system (1.10), it evidently satisfies the limiting system (1.6), that is (2.1) To locate the precise asymptotic behavior of blowing-up steady states to the limiting system (2.1) as d 2 → a 2 /λ 1 , we convert the blowing-up solutions of (2.1) to the bounded solutions by some suitable transformation.
For each fixed small ε = a2 λ1 − d 2 > 0, let (τ, w(x)) be a positive steady state of system (2.1), and denotẽ Multiplying the first equation of (2.1) by τ , then (τ ,w(x)) satisfies the system Let ε = 0, then system (2.3) becomes the following reduced system Solving the boundary problem ofw(x) in (2.4), we havẽ with constant µ = 0 to be determined later. Substituting (2.5) into the first equation of (2.4), then solving the limiting system (2.4) is deduced to finding a constant µ satisfying In one dimensional case e.g. Ω = (0, 1), by some simple computation it can be proved that g(µ) = A B has exactly two roots µ = ± 1 − B A , which also proves that the limiting system (2.4) has precisely two families of positive solutionsτ (1, l ± (x)) is a positive decreasing function in (0, 1) and l − (x) is a positive increasing function.
For multidimensional domain Ω, the existence of the roots of g(µ) is deeply investigated and proved in [12], we restate some basic estimates as follows.
Remark 1. In [12] it is proved that µg (µ) > 0 for µ = 0, due to the fact that Here it should be noted that .
(2.8) The existence of positive functions l(x) = b2 a2 (1+µ ± ϕ 1 (x)) and the nonzero of g (µ ± ) will be useful in our later investigation of existence and stability of the nontrivial positive steady states.
DenoteL a 2 λ 1 be the linearized operator of (2.4) around (k, kl(x)) with it can be checked later that zero is an eigenvalue ofL a 2 λ 1 for any k > 0 with an eigenvector (1, l(x)).
In the following, we shall prove that for small a2 λ1 − d 2 > 0 there exist positive nontrivial steady states of (2.3) perturbed from (k 0 , k 0 l(x)) for some specific positive constant k 0 > 0. Before proving the existence of the positive steady states of (2.3) for small a2 λ1 −d 2 > 0, we shall firstly prove that the zero eigenvalue ofL a 2 λ 1 is simple, such that the Lyapnov-Schmidt reduction method can be applied in our later proof of existence and stability analysis. : respectively, then zero is a simple eigenvalue of L a 2 Proof. Firstly, we prove that zero is an eigenvalue, where the eigenfunction (ξ, ψ(x)) satisfies (2.9) Solving the boundary value problem in (2.9), we have Substituting (2.10) into the first equation of (2.9) yields dx.
Note that f (K) is linear in K and by Lemma 2.3 and (2.8), we have with l(x) = b2 a2 (1 + µ ± ϕ 1 (x)) resp. and f (µ ± ) which proves that f (K) = 0 has precisely one root K = µ ± resp. for l(x) = b2 a2 (1 + µ ± ϕ 1 (x)) resp. Hence, zero is an eigenvalue of L a 2 In the following of this subsection, we shall focus on the existence of the bounded positive solution (τ ,w(x)) perturbed from some (k, kl(x)) for system (2.3), which corresponds to the existence of two types of solutions (τ, w(x)) to the limiting system (2.1) for small ε > 0. Lemma 2.4 and (2.12) imply that X = R×H 2 ν , and Y = R×L 2 have the following direct decomposition with ⊕ a direct sum in Y . Thus we can define a projection Q on Y satisfying ).
Remark 2. For one dimensional case e.g. Ω = (0, 1), by applying different transformation and by applying local and global bifurcation argument, we can further prove that the two local branches of positive monotone steady states stated in Theorem 1.1 can be extended to two global branches of positive monotone steady states for any d 2 ∈ (0, a2 λ1 ). The results on the global branches of monotone positive steady states and the detailed proof will be given in our forthcoming paper.
It is worth mentioning that in one or multidimensional case when d 2 → a 2 /λ 1 the asymptotic behavior of the ratio τ /w(x) of the large steady states to the limiting system (1.6) is the same as that of ξ/ψ(x) of the small steady states to the first limiting system (1.4). However even in one dimensional case it is still unclear that whether the steady states on the global branch tend to the steady state having spike layer or having other types of singular structure as d 2 → 0. It was proved in [9] that for the case A > B and small d 2 > 0 the SKT model with cross diffusion has a small spiky steady state near (a 1 /b 1 , 0), which is perturbed from the spiky steady state of the limiting system (1.6), and it is known that such spiky steady state is unstable [17].

2.2.
Spectral instability of nontrivial positive steady states to the limiting system. In this subsection, we shall investigate the spectral stability of the positive steady states (τ d2 , w d2 (x)) to the limiting system (1.10), which were constructed in previous subsection.
Proof. The proof of this Lemma is divided into three steps.
Firstly, we prove that a 2 is not an eigenvalue of system (2.26). By contradiction, suppose λ = a 2 is an eigenvalue of system (2.26) with an eigenfunction (ξ * , ψ * (x)) satisfying On the other hand, according to the Hölder inequality and the definition of µ ± stated in Lemma 2.3, we have This is contrary to (2.29), which proves that a 2 is not an eigenvalue of (2.26). Next, by applying similar argument as in the proof of Lemma 2.4, we can prove that zero is an eigenvalue of (H −1 a2/λ1 )L a2/λ1 with the eigenspace spanned by Ψ = (1, l(x)) , here we omit the detailed proof. Furthermore, it can be checked that H a2/λ1 Ψ / ∈ Range(L a2/λ1 ), which means that zero is also a simple eigenvalue of (H −1 a2/λ1 )L a2/λ1 . Finally, it remains to prove that (2.26) has no eigenvalue with nonnegative real part. Suppose that there exists an eigenvalue Reλ * ≥ 0 and λ * = a 2 , with the associated eigenfunction denoted by (ξ * , ψ * (x)), in such case the boundary value problem of ψ * (x) in (2.26) has a solution if and only if (2.31) Substituting (2.31) into the first equation of (2.26), it yields which can be reduced to the following algebraic equation of λ * The above inequality follows from g(µ ± ) = A B > 1 and the Cauchy-Schwartz inequality. Since E > 0, we can see that Reλ * < 0 from (2.33). This leads to a contradiction which completes the proof.
By Lemma 2.6 and the spectral perturbation argument, it is suffices to investigate the location of the eigenvalue for the problem (2.25) near zero when a2 λ1 − d 2 > 0 is small. This is the key point in the stability analysis of the steady states to the limiting system (1.10).
3. Existence and instability of nontrivial positive steady states to the original cross-diffusion system. In this section, we investigate the existence and stability of the nontrivial positive steady states to the original cross-diffusion system (1.3) when both d 1 and α = ρ12 d1 are large enough. Let r = 1 α , s = 1 d1 , w = αv and φ = (1 + αv)u, then the original cross-diffusion system (1.3) can be written as the following system To prove the existence of the steady state for the cross-diffusion system (1.3) when both d 1 and ρ12 d1 are large enough, it is equivalent to prove the corresponding existence result for system (3.1) when both r and s are small enough.

3.2.
Instability of nontrivial positive steady states to the original cross diffusion system. For each fixed small a2 λ1 − d 2 > 0 and small enough s, r > 0, let (φ s,r (x, d 2 ), w s,r (x, d 2 )) be one of the two types of nontrivial positive steady states of system (3.1) with φ s,r (x, d 2 ) = τ s,r (d 2 ) +φ s,r (x, d 2 ) obtained in Theorem 3.1.