Study On Boundary Blow-up Solutions For Elliptic Partial Differential Equations

In this paper,we discuss boundary blow-up solutions of elliptic problems which arise in the study of geometry,physics,chemistry,biology and other fields.Hence it is meaningful to research these problems.In the view of mathematics,the research concerning these problems mainly focus on the existence,uniqueness,multiplicity,boundary behavior of solutions and so on.Based on many previous works to these problems,in Chapter 2,we discuss boundary blow-up solutions for a homogeneous quasilinear elliptic problem with nonnegative weight where ?(?)Rn(n?1)is a smooth bounded domain.we mainly obtain the existence,uniqueness and boundary behavior of solutions.The next discussion bases on the results of this chapter.In Chapter 3,we study a semilinear elliptic boundary blow-up problem with sign-changing weights where ?(?)Rn(n?1)is a bounded smooth domain,(?)>0 is a parameter and a(?)= a+-(?)a_is the weight function.We verify that there exists(?)*>0 such that the problem has the minimal positive large solution for any(?)?(0,(?)*),while no positive large solutions for any(?)>(?)*.Then,by applying the local bifurcation theory,we analysis the structure of the branch produced by the positive solutions.In Chapter 4,we focus on a nonhomogeneous quasilinear elliptic boundary blow-up problem where p>1,? is a smooth bounded domain in Rn and h? C(?),We mainly analyze the influences caused by the increase of h near the boundary to the existence of solutions and obtain a optimal assumption of h such that the nonhomogeneous problem has boundary blow-up solutions.Furthermore,it can be verified that any boundary blow-up solution is nonnegative under an additional assumption.