| In this dissertation, we first prove a Hardy type inequality for u ∈ Wm,10 (O), where O is a bounded smooth domain in RN and m ≥ 2. For all j ≥ 0, 1 ≤ k ≤ m -- 1, such that 1 ≤ j + k ≤ m, it holds that 6jux dxm-j-k ∈ Wk,10 (O), where d is a smooth positive function which coincides with dist(x, ∂O) near ∂O, and ∂l denotes any partial differential operator of order l.;We also study a singular Sturm-Liouville equation --(x 2alphau')' + u = f on (0, 1), with the boundary condition u(1) = 0. Here alpha > 0 and f ∈ L2(0, 1). We prescribe appropriate (weighted) homogeneous and non-homogeneous boundary conditions at 0 and prove the existence and uniqueness of H2loc (0, 1] solutions. We study the regularity at the origin of such solutions. We perform a spectral analysis of the differential operator L u := --(x2alpha u')' + u under homogeneous boundary conditions.;Finally, we are interested in the equation --(|x| 2alphau')' + |u|p--1 u = micro on (--1, 1) with boundary condition u(--1) = u(1) = 0. Here alpha > 0, p ≥ 1 and micro is a bounded Radon measure on the interval (--1, 1). We identify an appropriate concept of solution for this equation, and we establish some existence and uniqueness results. We examine the limiting behavior of three approximation schemes. The isolated singularity at 0 is also investigated. |
