Positive solutions of a Lotka-Volterra competition model with cross-diffusion

and Applied Analysis 3 satisfy the relation (u, V, w, z) ≥ (u, V, w, z), and for all (u, V) ∈ U × V, satisfy the following inequalities: −Δw +M 1 w ≥ f 1 ( u, V) + M1A (u, V) , x ∈ Ω, −Δz +M 2 z ≥ f 2 ( u, V) + M2B (u, V) , x ∈ Ω, −Δw +M 1 w ≤ f 1 (u, V) + M 1 A (u, V) , x ∈ Ω, −Δz +M 2 z ≤ f 2 (u, V) + M 2 B (u, V) , x ∈ Ω, u ≥ A ∗ (w, z) , V ≥ B ∗ (w, z) , x ∈ Ω, u ≤ A ∗ (w, z) , V ≤ B ∗ (w, z) , x ∈ Ω, w ≥ 0 ≥ w, z ≥ 0 ≥ z, x ∈ ∂Ω. (11) We can have the following conclusion from [4, Theorem 2.1]. Proposition 4. Assume that (8) has coupled upper and lower solutions ((u, V, w, z), (u, V, w, z)), then there exists at least one solution (u, V, w, z), satisfying the relation (u, V, w, z) ≤ (u, V, w, z) ≤ (u, V, w, z) . (12) Furthermore, (u, V) is the solution of (5). Next, if u, V, u, V satisfy u = A ∗ (w, z) , V = B ∗ (w, z) , u = A ∗ (w, z) , V = B ∗ (w, z) , (13) then w = A (u, V) , z = B (u, V) , w = A (u, V) , z = B (u, V) , (14) (11) can be rewritten as −ΔA (u, V) + M 1 A (u, V) ≥ f 1 ( u, V) + M1A (u, V) , x ∈ Ω, −ΔB (u, V) + M 2 B (u, V) ≥ f 2 ( u, V) + M2B (u, V) , x ∈ Ω, −ΔA (u, V) + M 1 A (u, V) ≤ f 1 (u, V) + M 1 A (u, V) , x ∈ Ω, −ΔB (u, V) + M 2 B (u, V) ≤ f 2 (u, V) + M 2 B (u, V) , x ∈ Ω, A (u, V) ≥ 0 ≥ A (u, V) , B (u, V) ≥ 0 ≥ B (u, V) , x ∈ ∂Ω. (15) Synthetically, we have the following result. Theorem 5. If there is a pair of functions ((u, V), (u, V)), satisfying (u, V, A (u, V) , B (u, V)) ≥ (u, V, A (u, V) , B (u, V)) , (16) and for all (u, V) ∈ U × V, (15) is satisfied, then (5) has at least one solution (u, V), satisfying the relation (u, V) ≤ (u, V) ≤ (u, V). To make sure the upper and lower solutions reasonable, we give the following two lemmas; more details can be found in [8, 9]. Lemma 6. If the functions u, V ∈ C(Ω) satisfy u| ∂Ω = V| ∂Ω = 0, u| Ω > 0, (∂u/∂ν)| ∂Ω < 0, ν is the outer unit normal vector of ∂Ω, then there exists positive constant ε, such that u(x) > εV(x), for all x ∈ Ω. For the equation: −Δu = u (a − u) , x ∈ Ω, u = 0, x ∈ ∂Ω. (17) Lemma 7. If a > λ 1 , then (17) has a unique positive solution θ a satisfying θ a ≤ a. In addition, θ a is increasing with respect to a. 3. A Lotka-Volterra Competition Model with Two Cross-Diffusions In this section, the existence of positive solutions of (1) corresponding to α ≥ 0, β ≥ 0, is investigated by applying Theorem 5 to proveTheorem 1. Proof. We seek some positive constants R,K, δ, R, K > λ 1 sufficiently large and δ sufficiently small, Lemma 6 may guarantee the existence of θ R and θ K . It can be easily known from Hopf boundary lemma: