Some Results Of Positive Solutions For Elliptic Equation With Nonlocal Terms And Predator-prey Model With Cross-diffusion Coefficient

In this paper,we investigate a elliptic equation with nonlocal terms and a predatorprey model with cross-diffusion coefficient.We mainly use the local and global bifurcation theory,Lyapunov-Schmidt reduction technique and Whyburn technique in the elliptic equation with nonlocal terms;otherwise,we use the priori estimates of solutions,the fixed point index theory on cones,the maximum principle and the Harnack estimate in the predator-prey model with cross-diffusion coefficient.The paper consists of three chapters:In Chapter 1,we briefly introduce the background of this paper and give some necessary basic knowledge.In Chapter 2,we consider the elliptic problem with nonlocal terms Where Ω is a bounded domain of RN with a smooth boundary,N≥2;n is the outward unit normal to ?Ω;m(x),h(x)∈ Cθ(Ω),θ∈(0,1)and m(x),h(x)change sign in Ω;p>1 and q>1 are integer;λ∈R is a parameter;α=0,β=1 or α= 1,β=0 orα=β=1.In this chapter,we consider local and global bifurcation nature of positive solutions for ∫Ωm(x)dx≠0 and ∫Ωm(x)dx=0 in three cases α=0,α=1 and α=1,β=0 and α=β=1 by using the local and global bifurcation theory,Lyapunov-Schmidt reduction technique,property of principal eigenvalue,blow up technique and Whyburn technique.The global bifurcation structure of positive solutions set are obtained,Moreover,the main results of this chapter extend the related results on local elliptic boundary problem to the non-local elliptic problem,and we extend the methods obtaining a priori bounds for local elliptic equation to the non-local problem.In Chapter 3,we consider the following predator-prey model with cross-diffusion coefficient Where Ω is a bounded domain of RN with a smooth boundary,N=2,3;n is the outward unit normal to ?Ω a,b,m are positive constants;ρ(x)>0 is positive smooth function in Ω with ?ρ/?n=0 on k≥0 is the cross-diffusion coefficient.In this chapter,we consider the existence and the local bifurcation nature of positive solutions for the equation by using the priori estimates of solutions,the fixed point index theory on cones,the maximum principle,the Harnack estimate and the local bifurcation theory,etc.