Positive Solutions Of Quasilinear Elliptic Systems

Some quasilinear elliptic systems are investigated in this paper. The main results are existence, non-existence, multiplicity, and qualitative analysis of positive solutions. In Chapter 2, we consider the radial system of p-Laplacian. Using some more precise a priori estimates and topology theory, we obtain the existence of positive solutions.In Chapter 3, we discuss the qusilinear elliptic systems with the homogeneous Dirich-let boundary condition, and get the existence and multiplicity of positive solutions under the proper region of parameters. We try to show the existence results with variation method since the system has variation structure. However, when a(x), b(x), c(x) change their signs, the corresponding Eulur's function is unbounded from below on the space W₀^1,p(Ω) × W_<sup>1,q(Ω), which is difficult to use the variation method directly and get solution in W₀^1,p× W₀^1,q. But we can prove that the corresponding Eulur's function is bounded from below in a proper set S of W₀^1,p × W₀^1,q, and then a minimizer of Eulur's function in S(if it exists) gives rise to a solution of the problem. Precisely, exploiting the relationship between fibreing maps and S (by analysis, the solutions of the problem must lie in S), we discuss how S and subsets of S change as parameters λ,μ change and show that how the existence and non-existence results for positive solutions and linked to properties of S.In Chapter 4, we consider the qusilinear elliptic systems with the homogeneous Neumann boundary condition, and obtain the existence and multiplicity of positive solutions. We showmore wide region of existence results than it in chapter three. In the last of this chapter, wediscuss a special case: and also obtain the existence result, whichis considered little in other's papers.Finally, we investigate the degenerate ecological models. Using decoupling method [9], the system can be reduced to an equivalent single equation with a nonlinear term. Then, bifurcation results follows. We obtain the region of coexistence states by a priori estimate and bifurcation thoery. When p = q, we show the non-existence, existence and structure of coexistence states.