An Iterative Method For The Constrained Least Squares Solution Of The Matrix Equations

In this paper some p-Laplace quasilinear systems are investigated using fiberingmethod, sub-super solutions and mountain lemma. The main results are existence,non-existence and uniqueness.In Chapter 2, we discuss the p-Laplace quasilinear systems with the homogeneousDirichlet boundary condition using fibering method, and get the existence of positivesolutions under the proper region of parameters. We try to show the existence of posi-tive solutions to energy function J with fibering method since the system has variationstructure. However, when c(x) changes the sign, the corresponding function J is un-bounded from below on the space W01 ,p1(Ω)×W01 ,p2(Ω)×W01 ,p3, which is di?cult touse the fibering method directly and get solution in W01 ,p1(Ω)×W01 ,p2(Ω)×W01 ,p3. Butwe can change the solution of the original problem into resolving the auxiliary problemMλ,μ,νby several lemmas of section 2.3. On the above fundamental, we analyze thein?uence on the solution of auxiliary problem as parametersλ,μ,νchange, and exploitthe relationship between parameters and the original problem. Furthermore, we canobtain the existence of positive solutions to the problem.In Chapter 3, we consider p-Laplace quasilinear systems with Dirichlet boundaryconditions, which originate from the degenerate ecological model. When the first eigen-valueλ1(?,a) < 0 andμ1(?,b) < 0 to eigenvalue problem with weight function a(x)and b(x), respectively, we obtain the existence and uniqueness of positive solutions bysub-super solutions and weak comparison principle. Meanwhile, whenλ1(?,a)≥0 andμ1(?,b)≥0, we also obtain the non-existence of positive solution.In Chapter 4, we investigate existence of positive solutions to p-Laplace qusilinearsystems using mountain pass lemma. We prove that the function J corresponding tothe systems satisfies the conditions of mountain pass lemma by precise calculation andget the existence of positive solution, which is considered few in others'papers.