We are concerned with the solvability of boundary value problems and related general properties of solutions of semilinear equations and of quasilinear elliptic equations with a variety of domains. In Chapter 2, we concentrate on the study of existence and uniqueness of positive radially symmetric solutions of the equation (*) with a variety of Dirichlet and Neumann boundary conditions in annular domains. Using Leray-Schauder degree theory, we establish some new existence results. In Chapter 3, we shall give a new description of the generalized degree theory. In Chapter 4, we prove that there is a strong maximum principle for A+theta when 1 0. In Chapter 5, the existence and uniqueness of positive radial solutions of the problem (***) on O=BR with Dirichlet condition are proved.