Neumann problems for a degenerate elliptic equation in \({\mathbb{R}}^ n\) with a parameter (Q1090844)

Let L be a degenerate elliptic operator defined by \[ L=(1-| x|^ 2)\sum^{n}_{j=1}(\partial /\partial x_ j)^ 2 + \tau \sum^{n}_{j=1}x_ j(\partial /\partial x_ j), \] where \(\tau\) is a real parameter. We consider the boundary value problem \[ Lu=0 \text{ in D; Neumann type boundary condition on }\partial D. \] Then we can obtain various existence and uniqueness theorems corresponding to domain D and the parameter \(\tau\). The cases treated here are \((i)\tau >2\). (1) D is the unit ball. (2) D is an annular domain bounded by two concentric spheres, one of which is the unit sphere. (3) D is a simple connected region which contains the unit ball. (4) D is bounded by two concentric spheres and contains the unit sphere. (5) D is the same as (4) but different boundary conditions. (ii) \(\tau <-2\). (1) D is the same as (2) in (i). (2) D is the same as (1) but different boundary conditions. (3) D is the same as (4) in (i), various boundary conditions. (iii) \(-2<\tau <2\). (1) D is the same as (2) in (i). Or (2) D is the same as (4) in (i).