Liouville Theorem And Green Function For Degenerate Elliptic Equations

In my dissertation, I firstly study outside boundary value problem in exterior domain for a class of so called A-harmonic type equations, the Liouville's theorem of weak solutions is derived by use of the fundamental solution A—harmonic type equations and comparison lemma. Secondly, we prove the Maximum principle for inhomogeneous degenerate equations, some apriori esti-mates of nonlinear degenerate equations with X-elliptic conditions in the sense of distribution are established. By researching the modified Green function, the comparison of the Green functions with the fundamental solution of p-Laplacian equation on a Carnot group is derived. This will provide a new idea and method for researching the properties of X-elliptic operator. At last, According to the estimates of the Green function and the hole-filling technique, we will establish locally Holder continuity of weak solutions of X-elliptic equations with bounded measurable coefficients. Instead of classical De Giorgi-Moser-Nash iterating technique of equations with discontinuous coefficients, we here obtain that weak soluiton must satisfy the assumption of Morrey's Lemma by making use of Green function as a kernel function and the so-called hole-filling argument.