Nonnegative solutions for a heterogeneous degenerate competition model

NON-NEGATIVE SOLUTIONS FOR A HETEROGENEOUS DEGENERATE COMPETITION MODEL ANTONIO SU´ AREZ1 (March 30, 2004) Abstract This paper deals with the existence, uniqueness and qualitative properties of nonnegative and nontrivial solutions of a spatially heterogeneous LotkaVolterra competition model with nonlinear diffusion. We give conditions in terms of the coefficients involved in the setting of the problem which assure the existence of nonnegative solutions as well as uniqueness of positive solution. In order to obtain the results we employ monotonicity methods, singular spectral theory and a fixed point index. Short title: Degenerate competition problem 1. Introduction In this work we are mainly concerned with the existence and uniqueness of nonnegative solutions for the problem      L1(wm) = w(λ−a(x)w−b(x)z) in Ω, L2(zn) = z(µ−d(x)z−c(x)w) in Ω, w=z= 0 on ∂Ω, (1) where Ω is a bounded domain of IRNwith regular boundary ∂Ω, Lk,k= 1,2 are two second order uniformly elliptic operators of the form Lk:= − N X i,j=1 ak ij(x)DiDj+ N X i=1 bk i(x)Dik= 1,2,(2) 1Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Fac. Matem´aticas, C/ Tarfia s/n, C.P. 41012, Univ. Sevilla, Spain, e-mail: [email protected] c °Australian Mathematical Society 0, Serial-fee code 0334-2700/0 1 with ak ij, bk i∈C1(Ω); m, n > 1; λ, µ ∈IR and a, b, c, d ∈C1(Ω) nonnegative and nontrivial. Problem (1) provides us with the steady-state solutions to a related evolutionary problem, which models the behaviour of two competing species, with populations densities w(x) and z(x), inhabiting Ω. We refer to [14] for the meaning of each coefficient and details about the model. When m=n= 1 (linear diffusion), (1) has been extensively studied in the recent years. In the case that a, b, c and dare strictly positive functions, see for example [6], [7], [8], [10], [12], [13], [19], [20], [21], [25], [28], [32] and the references therein. When band/or cvanish in a domain of Ω (that means that, for instance, zdoes not interact with win the set B0:= {x∈ Ω : b(x)=0}); problem (1) was studied in [22], [26] and [28]. And finally, recently the case avanishes in a part of Ω but all other coefficients functions are strictly positive over Ω has been analysed in [18] and [27], where essential qualitative changes occur. Observe that in this case positive constants are not supersolutions of (1) and, in fact, it is shown that the a priori bounds are lost for some values of λand µappearing a new kind of positive solutions (which are infinite over a region of Ω and finite on the rest of Ω) that govern the behaviour of a related evolutionary problem. However, model (1) is less known when m, n > 1, and it has been only analysed under more restrictive hypotheses, with constant coefficients (homogeneous environmental case) in [14] and when aand dare strictly positive in [9] and [31], all of them with L1=L2=−∆. These new parameters (m, n) were introduced in [23] and [29] by describing the dynamics of biological population whose mobility depends upon their density. In this context, it means that the diffusion, the rate of movement of the species from high density regions to low ones, is slower than in the linear case, giving more realistic results. Mathematically, this has mainly three consequences which distinguish this system from the one with m=n= 1: the strong maximum principle does not apply (and so, unlike the linear case, there can exist nonnegative and nontrivial solutions which are not positive in all Ω), a-priori bounds for all the solutions of (1) and for all the values of λand µ, even when aor dvanishes, exist and that the linearized method cannot be applied directly. In order to study (1) we make the appropriate change of variables wm=u and zn=v, which transforms (1) into      L1u=u1/m(λ−a(x)u1/m −b(x)v1/n) in Ω, L2v=v1/n(µ−d(x)v1/n −c(x)u1/m) in Ω, u=v= 0 on ∂Ω. (3) Since only nonnegative solutions have physical interest, there are four types of solutions: the trivial one, the semitrivial solutions (u, 0) and (0, v), those with both components strictly positive, the coexistence states, and those 2 where at least one component could vanish in a part of Ω, the semicoexistence states. Observe that a semicoexistence state could be a coexistence one (see Proposition 3.3). Sometimes, we are able to prove that a semicoexistence state vanishes in a region of Ω (see Theorem 3.4), and so it is not a coexistence state. Now we describe the parts of this work stating their main results. Observe that the semitrivial solutions satisfy the following equation, the reason for our study in Section 2, (Lw =f(x)w1/r −g(x)w2/r in Ω, w= 0 on ∂Ω,(4) where Lis an operator of the form (2), f, g ∈C1(Ω) with g≥0, g6≡ 0, f can change sign and r=mor n. Although the semitrivial solutions give f≡λ(or µ) and so constant, it will be very useful to study (4) when f changes sign. This equation has been previously studied in [3], [14], [15], [24] and [30] assuming more restrictions in the data of (4). We collect the main results of these works, and as a consequence we obtain that the semitrivial solution (u, 0) = (resp. (0, v)) exists and it is unique if, and only if, λ > 0 (resp. µ > 0). Then, we study the existence of dead cores (see [17]) of the solutions of (4). Given a solution wof (4); we call the set Ω0:= {x∈Ω : w(x) = 0}, if this is nonempty, a dead core of w. We demonstrate a result which assures the existence of a dead core for any nonnegative solution of (4) under suitable hypotheses (see Theorem 2.4). A direct consequence of our result is that any nonnegative solution of (4) has dead core if the maximum of fis small. To our knowledge, the above results concerning to the existence of dead core have been obtained when L=−∆, see [3], [14], [17] and [30], with their proofs being based on the radial properties of the Laplacian. In this way our result generalises previous ones. In Section 3 we carry out an analysis of the existence of semicoexistence, coexistence states and dead cores of the system (3). Using the results of Section 2 and monotonicity methods we obtain results which can be summarized as follows: take λ∈IR, •Assume λ≤0: if µ∈(−∞,0] only the trivial solution exists, if µ∈(0,∞) only the trivial and the semitrivial solutions (0, v) exist; •Assume λ > 0: there exist positive values µ∗(λ), µ∗(λ), µ1(λ), µ2(λ) with µ1(λ)<min{µ∗(λ), µ∗(λ)}and µ2(λ)>max{µ∗(λ), µ∗(λ)} such that –If µ∈(−∞,0] only the trivial and semitrivial solution (u, 0) exist; 3 –If µ∈(0, µ1(λ)) there exists at least a semicoexistence state (u, v) and the component vhas dead core; –If µ∈(µ1(λ), µ2(λ)) there exists at least a semicoexistence state; –If µ∈(µ2(λ),∞) there exists at least a semicoexistence state (u, v) and the component uhas dead core; –If, moreover µ∗(λ)< µ∗(λ), then if µ∈(µ∗(λ), µ∗(λ)) there exists at least a coexistence state. Analogous results can be obtained when we fix the parameter µ. It’s worth mentioning that the existence of µ1(λ)>0 was shown in [14] when all the coefficients were positive constants. To our knowledge, the existence of µ2(λ)>0 is new. In Remark 3.1 we give a biological interpretation of this result. In Sections 4 and 5 we study the uniqueness of coexistence states of (3). For that we use the fixed point index. Observe that because m, n > 1 the linearization of (3) around the trivial or semitrivial solutions do not exist, so we cannot apply the results in [11] (see also [25] and [28]) to compute their indices. So, we will build appropriate homotopies for that. To compute the index of a coexistence state we can use a linearization. In this case the linearization of (3) around a coexistence state leads us to a eigenvalue problem of the form (LU+MU =σU in Ω, U= 0 on ∂Ω, (5) where L= diag(L1, L2) and M= (mij), 1 ≤i, j ≤2 with mij ≥0 for i6=jand mij blowing up near ∂Ω in a controlled way. Following [16] and [28] we define a specific order and establish the existence of the principal eigenvalue of (5) as well as a characterization of its positivity by means the existence of a supersolution. Now, we prove that, again with fixed λ > 0, there exists a unique coexistence state when µbelongs to a subset of (µ∗(λ), µ∗(λ)). Furthermore, if m=nand a, d are strictly positive functions we have uniqueness of coexistence state if bMor cMis small. The results about uniqueness of coexistence state of (3) are also, we believe, new. 2. Preliminaries. The degenerate logistic equation We consider the Banach space X:= C1 0(Ω) ordered by its cone of nonnegative functions P, whose interior is int (P) := {u∈X:u(x)>0 for all x∈Ω and ∂u/∂n < 0 on ∂Ω}, where ndenotes the outward unit normal on ∂Ω. We say that u∈Xis nonnegative, u≥0, if u∈P, and uis positive, u > 0, if u∈int (P). 4 Given q∈L∞(Ω) and Lan operator of the form (2), we denote by σ1(L+q) the principal eigenvalue of L+qsubject to homogeneous Dirichlet boundary conditions. Moreover, if we denote by ϕ∈int Pthe unique positive eigenfunction associated with σ1(L+q) normalized such that kϕk∞= 1, then it is well known that ∂ϕ ∂ν <0 on ∂Ω, (6) for νany direction out of Ω. Recall that as positive constants are supersolutions of L, then σ1(L)>0.(7) Finally, for f∈Y:= C0(Ω) we write fM:= max x∈Ω f(x), fL:= min x∈Ω f(x). 2.1. Existence of solutions In this section we study the semitrivial solutions of (3). Observe that if the solutions of (3) are of the form (u, 0) and (0, v), then satisfy equations of the following type (Lw =f(x)wq−g(x)wpin Ω, w= 0 on ∂Ω,(8) where Lis an operator of the form (2), f, g ∈C1(Ω) with g≥0, g6≡ 0, f can change sign and qand psatisfy (H) 0 < q < 1, p > q. Our first result gives us the existence of nonnegative solution of (8) and lists some useful properties. For a proof of this result see [15] for instance. Theorem 2.1. Assume (H). The following assertions are true: 1. There exists a maximal nonnegative and nontrivial solution of (8) if, and only if, fM>0. We denote it by θ[L,q,p,f,g]. 2. The following estimates hold: θ[L,q,p,f,g](x)≤f1/(1−q) Meq/(1−q) Me(x)x∈Ω, (θ[L,q,p,f,g])M≤(fMeM)1/(1−q),(9) where e∈C2(Ω) is the unique solution of (Le = 1 in Ω, e= 0 on ∂Ω.(10) 3. If w∈C1(Ω) is a nonnegative subsolution of (8), then w≤θ[L,q,p,f,g]. 5 4. Let fi∈C1(Ω),i= 1,2be such that f1≤f2, then θ[L,q,p,f1,g]≤ θ[L,q,p,f2,g]. 5. If fL>0, then any nonnegative solution of (8) is positive. Moreover, in this case there exists a unique positive solution and it satisfies εϕ(x)≤θ[L,q,p,f,g](x)x∈Ω,(11) where εis the unique positive root of σ1(L)ε1−q+gMεp−q=fL.(12) Remark. If we consider fLas a real parameter, then it is easy to prove that as fL→ ∞,ε(fL) = O(f1/(1−q) L) when p≤1 and ε(fL) = O(f1/(p−q) L) when p > 1. 2.2. Existence of dead cores In order to state and prove the main result, we need some preliminary ones. Lemma 2.2. Let R > 0and γ > 0. Consider the problem (Lw =−Rwq−g(x)wpin Ω, w=γon ∂Ω.(13) Then, there exists a unique nonnegative solution of (13). Proof. For the existence we use the sub-supersolution method. Indeed, it is easy to prove that (w, w) = (0, γ) is a sub-supersolution of (13). For the uniqueness we can apply Theorem 2 in [1]. The following technical result is fundamental in our study. Moreover, it generalizes Lemma 7 in [30] and Lemma 2.5 in [3], where a similar result was proved when L=−∆ and g(x)≡0. Lemma 2.3. We fix γ > 0and β > 2/(1 −q). Let δ0be such that for all x, x0∈IRNsuch that 0≤ |x−x0| ≤ δ0 |x−x0|βq +β|x−x0|β−1L(|x−x0|)+ +β(1 −β)|x−x0|β−2 N X i,j=1 aij(x)Di(|x−x0|)Dj(|x−x0|)≥0.(14) Then, for all 0< δ < dist(x0, ∂Ω), the unique nonnegative solution, w, of (13) in B(x0, δ)is such that w(x0) = 0 provided that R≥µγ min{δ, δ0}β¶1−q .(15) 6 Remark. Observe that since β > 2/(1−q), then βq < β−2< β−1, and so the existence of δ0satisfying (14) is guaranteed. Moreover, since β > 2 (14) can be considered in a classical sense. Proof. Consider the function Φ(x) := (Φ1(x) := R1/(1−q)|x−x0|βif x∈B(x0, δ0), Φ2(x) := R1/(1−q)δβ 0if x∈B(x0, δ)\B(x0, δ0), with Φ ≡Φ1if δ≤δ0. By the choice of β, we have that Φ1∈H2(B(x0, δ0)). Moreover, ∂Φ1 ∂nL ≥0,on ∂B(x0, δ0), where nLstands for the conormal associated with L, i.e., (nL)i:= PN j=1 aijnj. Indeed, for x∈∂B(x0, δ0) we have ∂Φ1 ∂nL (x) = R1/(1−q)β|x−x0|β−3( N X i,j=1 aij(x)(xi−xi 0)(xj−xj 0)) ≥0. Moreover, L(Φ1) + RΦq 1+g(x)Φp 1=R1/(1−q)(β|x−x0|β−1L(|x−x0|)+ +β(1 −β)|x−x0|β−2 N X i,j=1 aijDi(|x−x0|)Dj(|x−x0|))+ +RRq/(1−q)|x−x0|βq +g(x)Rp/(1−q)|x−x0|βp ≥ ≥R1/(1−q)(|x−x0|βq +β|x−x0|β−1L(|x−x0|)+ β(1 −β)|x−x0|β−2 N X i,j=1 aijDi(|x−x0|)Dj(|x−x0|)) ≥0, by (14). In B(x0, δ)\B(x0, δ0), we have that L(Φ2) + RΦq 2+g(x)Φp 2≥0. Finally, in ∂B(x0, δ), Φ is bigger than γprovided that (15) holds. Hence, we can apply Lemma I.1 in [4] and conclude that Φ is a supersolution of (13) in B(x0, δ). This completes the proof. For R > 0, we define the set N(R) := {x∈Ω : f−(x)≥R}={x∈Ω : f(x)≤ −R}, where f±(x) := max{±f(x),0}. Assume that f±6≡ 0. The main result of this section is the following one. Theorem 2.4. Assume that there exists R > 0such that 7 1. δR:= µfMeM R¶1/(β(1−q)) ≤δ0, 2. M(R) := {x∈N(R) : dist(x, ∂N(R)\∂Ω) ≥δR} 6=∅. Then, there exists a dead core for any nonnegative solution wof (8). Moreover, we have M(R)⊂Ω0={x∈Ω : w(x) = 0}. Proof. Let x0∈M(R), then B(x0, δR) := {x∈Ω : |x−x0|< δR} ⊂ N(R).(16) We call zthe unique nonnegative solution of (13) in B(x0, δR) with γ= (fMeM)1/(1−q). Then, by (16) we have that Lθ[L,q,p,f,g]≤ −Rθq [L,q,p,f,g]−g(x)θp [L,q,p,f,g]in B(x0, δR), which implies that L(z−θ[L,q,p,f,g])≥R(θq [L,q,p,f,g]−zq) + g(x)(θp [L,q,p,f,g]−zp) in B(x0, δR), and by (9) and the choice of γwe get z≥θ[L,q,p,f,g]on ∂B(x0, δR). Hence, if we denote by Ω1:= {x∈ B(x0, δR) : z(x)< θ[L,q,p,f,g](x)}then L(z−θ[L,q,p,f,g])≥0 in Ω1, z−θ[L,q,p,f,g]≥0 on ∂Ω1∩∂B(x0, δR), z−θ[L,q,p,f,g]= 0 on ∂Ω1∩ B(x0, δR). The maximum principle implies that z≥θ[L,q,p,f,g]in B(x0, δR). Finally, we can apply Lemma 2.3 because δRsatisfies (15). This finishes the proof. As consequence of the above result, we have Corollary 2.5. Any nonnegative solution of (8) has a dead core provided that fMis sufficiently small. Proof. It is sufficient to repeat the proof of Remark 2.13 in [14] and to take account that δR→0 as fM→0. 8 3. Existence of nonnegative solutions Hereafter we write θ[L1,f,g]:= θ[L1,1/m,2/m,f,g], θ[L2,f,g]:= θ[L2,1/n,2/n,f,g]. The following result gives us a necessary and sufficient condition to obtain semicoexistence states. Theorem 3.1. Problem (3) has a semicoexistence state if, and only if, λ > 0and µ > 0. Proof. By Theorem 2.1 3) it follows that u≤θ[L1,λ,a], v ≤θ[L2,µ,d].(17) So, if λ≤0, again by Theorem 2.1 1) we obtain that u≡0. Analogously, if µ≤0, v≡0. Assume now that λ > 0 and µ > 0. In this case, we have that A(x) := λ−b(x)θ1/n [L2,µ,d](x)B(x) := µ−c(x)θ1/m [L1,λ,a](x) (18) satisfy AM=λ > 0 and BM=µ > 0. We consider the pair (u, u) = (θ[L1,A,a], θ[L1,λ,a]),(v, v) = (θ[L2,B,d], θ[L2,µ,d]). By definition of Aand Band Theorem 2.1 it follows that u≤u,v≤v and that uand vare nonnegative and nontrivial functions. Finally, it is not hard to prove that the pair (u, u)−(v, v) is a sub-supersolution of (3). This completes the proof. The following result provides us with conditions which assure the existence of coexistence states as well as bounds of them. Theorem 3.2. If λand µsatisfy λ > (b(x)θ1/n [L2,µ,d])M, µ > (c(x)θ1/m [L1,λ,a])M,(19) then, (3) possesses a coexistence state. Moreover, for any coexistence state (u, v)of (3) we have the following estimates: if λ > (b(x)θ1/n [L2,µ,d])Mthen ε1ϕ1≤θ[L1,A,a]≤u≤θ[L1,λ,a]≤λm/(m−1)(e1)1/(m−1) Me1,(20) and if µ > (c(x)θ1/m [L1,λ,a])M, then ε2ϕ2≤θ[L2,B,d]≤v≤θ[L2,µ,d]≤µn/(n−1)(e2)1/(n−1) Me2,(21) 9 The following result will be used to compare principal eigenvalues of different matrices. Lemma 4.6. Let A(x)=(aij(x)) and B(x)=(bij(x)) be two matrices with aij, bij satisfying (HM), bii ≥aii and aij ≥bji for i6=jwith some inequality strict. Then, σ1(L+A)< σ1(L+B). Proof. Let ΦAÂ0 be the eigenfunction associated with L+A. Then, it is easy to show that (L+B−σ1(L+A)I)ΦAÂ0, and so, ΦAis a strict supersolution of L+B−σ1(L+A)I. Hence, by Theorem 4.2 we deduce that σ1(L+B−σ1(L+A)I)>0, whence the conclusion follows. 5. Uniqueness result Along this section we assume that λand µsatisfy (19), and so the validity of the strong maximum principle is guaranteed. Indeed, by (21) we get u1/m(λ−b(x)v1/n)−a(x)u2/m ≥u1/m(λ−b(x)θ1/n [L2,µ,d])−a(x)u2/m, and so, by (19), there exists a positive constant Msuch that u1/m(λ−b(x)v1/n)−a(x)u2/m +Mu ≥0,(32) whence it follows that if (u, v) is a non-negative solution of (3) with u6≡, then u(x)>0 for all x∈Ω. Similarly we can reason with the second equation in (3). In this section we obtain a uniqueness result for a coexistence state of (3). In order to get the result we use the fixed point index in cones. Fixed M > 0 obtained in (32), consider the operator K:X27→ X2 defined by K(u, v) := Ã(L1+M)−1(u1/m(λ−a(x)u1/m −b(x)v1/n) + Mu) (L2+M)−1(v1/n(µ−d(x)v1/n −c(x)u1/m) + Mv)!, where (Li+M)−1,i= 1,2, stands for the inverse on the operator Li+M in Ω under homogeneous Dirichlet boundary conditions. Observe that by (7), σ1(Li+M)>0 and so (Li+M)−1is well-defined and it is a compact operator. Thanks to the choice of M, see (32), Kis a positive operator whose fixed points are componentwise nonnegative solutions of (3). 16 On the other hand, by (20) and (21), there exist Ri>0, i= 1,2, such that for every (u, v) coexistence states of (3) kuk∞≤R1:= (λ(e1)M)m/(m−1),kvk∞≤R2:= (µ(e2)M)n/(n−1). So, the fixed point index of Kover Bwith respect to the cone P×Pis well defined, where B:= {(u, v)∈P2:kuk∞≤R1+ 1,kvk∞≤R2+ 1}. Now, we are going to compute this index in some cases. Proposition 5.1. Assume that λand µsatisfy (19). The following assertions are true: 1. iP×P(K,B) = 1; 2. iP×P(K,(0,0)) = 0; 3. iP×P(K,(θ[L1,λ,a],0)) = iP×P(K,(0, θ[L2,µ,d])) = 0. Proof. 1.) Firstly, we define G1:X7→ Xby G1(u) := (L1+M)−1(u1/m(λ−a(x)u1/m) + Mu) By (9), taking Bu:= {u∈P:kuk∞≤R1+ 1}the fixed point index of G1 over Buis well-defined. Applying Lemma 12.1 in [2] it can be proved that iP(G1, Bu) = 1.(33) Indeed, if there exist t≥1 and u∈Psuch that kuk∞=R1+ 1 and G1(u) = tu, then L1u≤u1/m(λ t−a(x) tu1/m), and so, kuk∞≤µλ t¶m/(m−1) (e1)m/(m−1) M≤R1< R1+ 1. Analogously, iP(G2, Bv) = 1 (34) with G2(v) := (L2+M)−1(v1/n(µ−d(x)v1/n) + Mv) and Bv:= {v∈P: kvk∞≤R2+ 1}. Consider the operator H1: [0,1] ×X27→ X2defined by H1(t, u, v) := Ã(L1+M)−1(u1/m(λ−a(x)u1/m −tb(x)v1/n) + Mu) (L2+M)−1(v1/n(µ−d(x)v1/n −tc(x)u1/m) + Mv)!. 17 Observe that by (20) and (21) any fixed point of H1belongs to B. So, it follows by homotopy invariance, (33) and (34) that iP×P(K,B) = iP×P(H1(1,·),B) = iP×P(H1(0,·),B) =iP(G1, Bu)·iP(G2, Bv) = 1. We now prove 2). Let ψi∈Y,i= 1,2, be such that ψi>0 in Ω. We define H2(t, u, v) := Ã(L1+M)−1(u1/m(λ−a(x)u1/m −b(x)v1/n) + Mu +tψ1) (L2+M)−1(v1/n(µ−d(x)v1/n −c(x)u1/m) + Mv +tψ2)!. We claim that there exists δ > 0 such that (u, v)6=H2(t, u, v),∀t∈[0,1],∀(u, v)∈ Nδ,(35) where Nδ:= {(u, v)∈P2:kuk∞≤δ, kvk∞≤δ}\{(0,0)}. Assume there exist sequences (ur, vr) of functions and tr∈[0,1] such that (ur, vr)→(0,0) as r→ ∞ and (ur, vr) = H2(tr, ur, vr). Since λ > 0 and kvrk∞→0, there exists r0∈IN such that (λ−b(x)v1/n r)L> 0 for r≥r0. So, the strong maximum principle is satisfied in the first equation, and so ur>0. Let K > 0 be such that K≥σ1(L1). Since kurk∞→0, there exists r1∈IN such that for r≥r1we have L1ur=u1/m r(λ−b(x)v1/n r)−a(x)u2/m r+trψ1> Kur, and hence σ1(L1−K)>0, a contradiction. Thus, by (35) the homotopy is admissible and we get iP×P(K,(0,0)) = iP×P(K,Nδ) = iP×P(H2(0,·),Nδ) =iP×P(H2(1,·),Nδ) = 0, this last equality follows by (35). It remains to prove 3). Let ψ∈Ybe such that ψ > 0 in Ω. We define another operator H3(t, u, v) := Ã(L1+M)−1(u1/m(λ−a(x)u1/m −b(x)v1/n) + Mu) (L2+M)−1(v1/n(µ−d(x)v1/n −c(x)u1/m) + Mv +tψ)!. We claim that there exists δ > 0 such that (u, v)6=H3(t, u, v),∀t∈[0,1],∀(u, v)∈ Mδ,(36) where Mδ:= {(u, v)∈P2:ku−θ[L1,λ,a]k∞≤δ, kvk∞≤δ} \ {(θ[L1,λ,a],0)}. Assume there exist sequences (ur, vr)→(θ[L1,λ,a],0) as r→ ∞ and tr∈[0,1] such that (ur, vr) = H3(tr, ur, vr). 18 Since ur≤θ[L1,λ,a], and µ > (c(x)θ1/m [L1,λ,a])Mit follows that µ−c(x)u1/m r(x)≥µ−c(x)θ1/m [L1,λ,a]>0.(37) Let K > 0 be such that K≥σ1(L2). Then, by (37) there exists r0∈IN such that for r≥r0we have L2vr=v1/n r(µ−c(x)u1/m r)−d(x)v2/n r+trψ > Kvr,in Ω, and hence σ1(L2−K)>0, a contradiction. Thus, by (36) the homotopy is admissible and we get iP×P(K,(θ[L1,λ,a],0)) = iP×P(K,Mδ) = iP×P(H3(0,·),Mδ) =iP×P(H3(1,·),Mδ) = 0. Analogously, it can be treated the solution (0, θ[L2,µ,d]). Now, let (u0, v0) be a coexistence state of (3). We consider the matrix M(u0,v0):= (mij), i, j = 1,2, which is related to the linearization of (3) about (u0, v0), where m11 =−1 mu1/m−1 0(λ−2a(x)u1/m 0−b(x)v1/n 0), m12 =1 nb(x)u1/m 0v1/n−1 0, m21 =1 mc(x)v1/n 0u1/m−1 0, m22 =−1 nv1/n−1 0(µ−2d(x)v1/n 0−c(x)u1/m 0). (38) Observe that since (u0, v0) is a coexistence state, by (20) and (21) there exists k0>0 such that k0dist(x, ∂Ω) ≤u0, k0dist(x, ∂Ω) ≤v0, then M(u0,v0)satisfies (HM), so that σ1(L+M(u0,v0)) makes sense. The general uniqueness result reads Theorem 5.2. Assume that λand µsatisfy (19) and σ1(L+M(u0,v0))> 0for any (u0, v0)coexistence state of (3). Then, (3) possesses a unique coexistence state. Proof. Recall that by Proposition 3.3, if λand µsatisfy (19) then any nonnegative solution of (3) is a coexistence state. We claim that if (u0, v0) is a coexistence state of (3), then iP×P(K,(u0, v0)) = 1.(39) 19 Assume that we have proved (39), then since Kis a compact operator, it possesses a finite number of coexistence states, say (ui, vi), i= 1, . . . , r. Then, iP×P(K,B) = iP×P(K,(0,0)) + iP×P(K,(θ[L1,λ,a],0)) +iP×P(K,(0, θ[L2,µ,d])) + r X i=1 iP×P(K,(ui, vi)) and so, by Proposition 5.1 and (39), 1 = 0 + 0 + 0 + r, whence the conclusion now easily follows. It remains to prove (39). Let h∈C1(Ω) be such that hverifies that |h(x)|dist(x, ∂Ω)2−α≤Kfor some α∈(0,2], K > 0 and h≥max{0, m11, m22},(40) where m11 and m22 are defined in (38). We define the operator T(u, v) := Ã(L1+h)−1(u1/m(λ−a(x)u1/m −b(x)v1/n) + hu) (L2+h)−1(v1/n(µ−d(x)v1/n −c(x)u1/m) + hv)!. Observe that (Li+h)−1exists because h≥0 and so σ1(Li+h)>0. By the Leray-Schauder formula, iP×P(T,(u0, v0)) = (−1)ξ, where ξis the sum of the multiplicities of the eigenvalues of D(u,v)T(u0, v0) larger than one, being D(u,v)T(u0, v0) the linearization of Tabout (u0, v0). It is clear that D(u,v)T(u0, v0) = diag((L1+h)−1,(L2+h)−1)(−M(u0,v0)+ diag(h, h)), where M(u0,v0)is defined by (38). It is not hard to prove that if r > 1 is an eigenvalue of D(u,v)T(u0, v0), then σ1(L+M(u0,v0)+B) = 0,(41) where B=Ã(m11 −h)(1 r−1) m12(1 r−1) m21(1 r−1) (m22 −h)(1 r−1) ! Since r > 1, by (40) and Lemma 4.6 we get σ1(L+M(u0,v0)+B)> σ1(L+M(u0,v0))>0, contradicting (41). The following result provides us with a sufficient condition for σ1(L+ M(u0,v0))>0 to be hold. 20 Proposition 5.3. Assume that m=n,a(x), d(x)>0for all x∈Ω,λ and µsatisfy (19) and that for any (u0, v0)coexistence state of (3) µb a¶Mµc d¶Mµu0 v0¶(2−m)/m Mµv0 u0¶(2−m)/m M <1.(42) Then, (3) possesses a unique coexistence state. Proof. Let Φ := (αu1/m 0,−βv1/m 0)∈(C2(Ω) ∩C0 0(Ω))2, with α, β > 0 to be chosen. We will show that Φ is a supersolution in the sense of Definition 4.1 of L+M(u0,v0)if (42) holds. Proposition 4.3 and Theorem 5.2 will complete the proof. Firstly, observe that Φ Â0. In order to show that Φ is a supersolution of L+M(u0,v0)we have to prove that (for the first equation) L1(αu1/m 0) + m11(x)αu1/m 0+m12(x)(−βv1/m 0)>0,(43) where m11 and m12 are defined in (38). Taking into account the fact that L1(u1/m 0) = 1 mu1/m−1 0[(1 −1 m)u−1 0 N X i,j=1 a1 ijDiu0Dju0+L1u0], to prove (43) it suffices that a(x)u2/m−1 0> b(x)v2/m−1 0·β α,for all x∈Ω. Analogously, for the second equation it is sufficient that d(x)v2/m−1 0> c(x)u2/m−1 0·α β,for all x∈Ω. Now, by (42) it is easy to show that there exist αand βsatisfying the above inequalities. The following result provides us another sufficient condition to obtain a uniqueness result. Proposition 5.4. Assume that λand µsatisfy (19) and that for any coexistence state (u0, v0)of (3) the following inequalities hold for all x∈Ω, λ(1 −1 m) + a(x)u1/m 0(x)( 2 m−1) > b(x)µ1 + 1 n−1 m¶v1/n 0(x), µ(1 −1 n) + d(x)v1/n 0(x)( 2 n−1) > c(x)µ1 + 1 m−1 n¶u1/m 0(x). (44) Then, (3) possesses a unique coexistence state. 21 Proof. Taking Φ := (u0,−v0), it suffices to prove that Φ Â0 is a supersolution of L+M(u0,v0)provided that (44) and apply again Proposition 4.3 and Theorem 5.2. For the second equation, Φ is a supersolution if L2(−v0) + m21(x)(u0) + m22(x)(−v0)<0, where m21 and m22 are defined in (38). For observe that L2(−v0) + m21(x)(u0) + m22(x)(−v0) =v1/n 0³µ³1 n−1´+d(x)v1/n 0³1−2 n´+c(x)u1/m 0³1 + 1 m−1 n´´<0, provided that (44) holds. Similarly we can reason with the first equation. Now, we will use the upper estimates of (20) and (21) giving sufficient conditions for the uniqueness of coexistence state in terms of several coefficients involved in the model setting. Corollary 5.5. Assume that m=n,a(x), d(x)>0for x∈Ω,λand µsatisfy (19) and Ã(e1)1/(m−1) M ε1 (e2)1/(m−1) M ε2! 2−m mµe1 ϕ2¶ 2−m m Mµe2 ϕ1¶ 2−m m M (λµ) 2−m m−1<aLdL bMcM , (45) where ε1and ε2are defined in (22). Then, (3) possesses a unique coexistence state. Proof. By (20) and (21) we have that µu0 v0¶M ≤λm/(m−1)(e1)1/(m−1) M ε2µe1 ϕ2¶M , and µv0 u0¶M ≤µm/(m−1)(e2)1/(m−1) M ε1µe2 ϕ1¶M , and so, (42) is satisfied if (45) holds. It suffices to apply Proposition 5.3. Corollary 5.6. Assume that some of the following sets of inequality, 1 to 4, holds: 1. If 1< m, n ≤2, bMµ1 + 1 n−1 m¶µ1/(n−1)(e2)1/(n−1) M< λ(1 −1 m), cMµ1 + 1 m−1 n¶λ1/(m−1)(e1)1/(m−1) M< µ(1 −1 n), 22 2. If 1< n ≤2and m > 2, bM³1 + 1 n−1 m´µ1/(n−1)(e2)1/(n−1) M+ (1 −2 m)aMλ1/(m−1)(e1)1/(m−1) M < λ(1 −1 m), cM³1 + 1 m−1 n´λ1/(m−1)(e1)1/(m−1) M< µ(1 −1 n), 3. If 1< m ≤2and n > 2, cM³1 + 1 m−1 n´λ1/(m−1)(e1)1/(m−1) M+ (1 −2 n)dMµ1/(n−1)(e2)1/(n−1) M < µ(1 −1 n), bM³1 + 1 n−1 m´µ1/(n−1)(e2)1/(n−1) M< λ(1 −1 m), 4. If m > 2and n > 2, bM³1 + 1 n−1 m´µ1/(n−1)(e2)1/(n−1) M+ (1 −2 m)aMλ1/(m−1)(e1)1/(m−1) M < λ(1 −1 m), cM³1 + 1 m−1 n´λ1/(m−1)(e1)1/(m−1) M+ (1 −2 n)dMµ1/(n−1)(e2)1/(n−1) M < µ(1 −1 n), then, (3) possesses a unique coexistence state. Proof. Reasoning as in the proof of Corollary 5.5, it is sufficient to apply (20), (21) and Proposition 5.4. Remark. 1. Observe that when m= 1, (42) is the condition obtained in Theorem 4.2 in [28] and Theorem 4.8 in [22]. Moreover, when m=n, and aand dare positive, we obtain uniqueness provided that bMor cMis small. 2. The (λ, µ)-regions defined in Corollary 5.6 are subsets of the coexistence region obtained in Theorem 3.2. Similar conditions to those imposed in Figure 1 assure the existence of these subregions. 6. Conclusions We have studied the set of non-negative solutions of a spatially heterogeneous Lotka-Volterra competition model with degenerate diffusion. Basically, we have found three differences with the respect to the non-degenerate (linear) case: 1. In the degenerate case all the non-negative solutions are bounded, unlike the linear case in which a-priori bounds are lost for some values of the data of the problem. 23 2. In the degenerate case a new kind of non-negative solutions appears: non-negative and nontrivial solutions that vanish in a region of the habitat of the species. We obtain sufficient conditions in terms of some parameters involved in the setting of the model ensuring the existence or non-existence of such kind of solutions. 3. Unlike the non-degenerate case, in our model when the competition between the species is “strong” neither species drives to the other to extinction. Finally, we have obtained uniqueness of positive solution of the problem under some conditions on the data of the problem. Acknowledgments: The author is in grateful to Professor M. Delgado for his helpful comments. 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