双曲空间上几类偏微分方程的研究

The program will study some elliptic equaitons and parabolic equations on hyperbolic space. For the elliptic equations, we mainly discuss the existence of the solution which is invariant under some group action for semilinear elliptic problems, the existence of infinity many solutions for critical equations on hyperbolic space, the asymptotic behavior of the ground state solutions and the classificaion of the solutions for Henon equation on hyperbolic space and the existence of solutions of some elliptic equations on bounded domain. For the parabolic equation, we hope to get the threshold result for some parabolic problem on hyperbolic space.Hyperbolic space is a simply connected non-compact Riemannian manifold with constant sectional curvature ?1. This contrasts with Euclidean Space, where the sectional curvatrue is zero. Although hyperbolic space with N-dimension is diffeomorphic to Euclidean space with N-dimension , its negative-curvature metric gives it very different geometric and topological properties. Through studying the above problems, we want to make a good understanding about the commonness and difference between the study of partial differential equations in hypebolic space and Euclidean space.

本项目拟讨论双曲空间上两类偏微分方程，一类是椭圆方程，一类是抛物方程。关于椭圆方程我们主要研究双曲空间上半线性椭圆方程在群作用下不变的解的存在性; 临界方程无穷多变号解的存在性;Henon方程基态解的性质及解的分类和有界区域上椭圆方程解的存在性等.关于抛物方程，我们希望得到抛物方程的门槛结果。双曲空间是单连通的非紧黎曼流形且其截面曲率为-1，而欧氏空间的曲率为0。虽然N-维双曲空间与N-维欧氏空间是微分同胚的，然而其负的截面曲率使得它有不同的几何与拓扑性质。 本项目希望通过对以上问题的研究，揭示双曲空间与欧氏空间在偏微分方程的研究中的共性与区别。

本项目主要讨论双曲空间上半线性椭圆问题解的存在性及性质。首先我们讨论了全双曲空间上Henon 方程正解的存在性；.其次研究了双曲空间的有界区域上具有临界指数增长的Henon 方程正解的存在性及Henon 方程基态解的存在性与渐近性质；紧接着我们研究了双曲空间上非线性方程组解的存在性、解的性质、双曲空间上Henon-Hardy 强不定方程组无穷多解的存在性及双曲空间上超临界椭圆问题解的正则性；最后我们讨论了带Robin 边值条件的Henon 方程基态解的存在性及其渐近行为。