Study Of Singular Elliptic Boundary-value Problem And Nonlocal Elliptic Boundary-value Problem

With the development of science technology, the singular elliptic boundary value problem and nonlocal elliptic boundary value problem have more wider ap-plication background and profound mathematical meaning, which have become an important concern of mathematicians and other scientists. This thesis studies the given singular elliptic boundary value problem and nonlocal elliptic boundary value problem, it is divided into three chapters.In the first chapter, we introduce the background and research on this field as well as give some definitions and basic lemmas.In the second chapter, we discuss the singular elliptic boundary value problem where ? is a bounded domain in RN, N? 2, with C2,? boundary (?)?,? > 0,? ?(0,1) and ? > 0 is a real parameter. Now we list some conditions for convenience:(fl) f:R+ ?R is a continuous function;(f2) s-1f(s) is strictly increasing for s > 0;(f3) f :R+? R is strictly increasing.In the second chapter, firstly, using the upper-lower solution method, we obtain that(1) has at least one positive solution, and if 0 < 7 < 1, (1) has one and only one positive solution. In the meanwhile,the boundary behavior of the positive solution is established for 0 < ? < 1. Finally, we obtain the asymptotic behavior of the positive solutions under a special form of f(u), which satisfies (fl)-(f3).In the third chapter, we consider the nonlocal elliptic boundary value problem where ? is a bounded domain in RN, N? 2, with C2,? boundary (?)?, b?R, p > 0,??(0,1) and ? > 0 is a real parameter. In the third chapter, firstly, using the bifurcation theory, we obtain the bifurcation result of the positive solutions. Next,we discuss a priori bounds of the positive solutions for b > 0 as well as non-existence results of (2). Finally, using the upper-lower solution method, we obtain that (2)has at least one positive solution, and if b ? 0, (2) has one and only one positive solution.