Qualitative studies of solutions to elliptic equations in unbounded domains

(1)椭圆方程的振动判据：建立了首项为简并拉普拉斯的二阶拟线性椭圆方程的有效振动判据。本文的方法基于拟线性常微分方程的比较原理和渐近理论。对于一维情况，通过广义Prufer变换，得到了解的0个数的有用信息。(2) BVPs的liouville型定理和正解的不存在性：建立了首项为退化拉普拉斯算子和（广义）平均曲率算子的拟线性椭圆方程的lioville型定理。我们的结果可以看作是经典刘维尔定理的自然推广。(3)椭圆型问题正解的对称性：利用运动平面法或运动球面法证明了某些类型椭圆型方程的正解是径向对称的。从我们的结果中可以得到抛物型问题自相似解的有用信息。(4)双参数特征值问题：考虑双参数非线性Sturm-Liouville问题。证明了变分特征值的存在性。得到了特征值和特征函数的渐近公式。(5)椭圆系统解的渐近理论：研究半线性椭圆系统。建立了正解的不存在性判据、liouville型定理和振荡判据。

(1) Oscillation criteria for elliptic equations : Effective oscillation criteria are established for second-order quasilinear elliptic equations whose leading terms are degenerate Laplacians. Our method is based on comparison principles and asymptotic theory for quasilinear ordinary differential equations. For one-dimensional case, useful information about numbers of zeros of solutions is obtained via the generalized Prufer transformation.(2) Liouville-type theorems and nonexistence of positive solutions of BVPs : Lioville-type theorems are established for quasilinear elliptic equations whose leading terms are degenerate Laplacians and (generalized) mean curvature operators. Our results can be regarded as a natural extension of classical Liouville theorem.(3) Symmetry of positive solutions for elliptic problems : We prove by means of the moving plane method or moving sphere method that positive solutions of elliptic equations of certain types are radially symmetric. Useful information about self-similar solutions of parabolic problems can be derived from our results.(4) Two-parameter eigenvalue problems : Two-parameter nonlinear Sturm-Liouville problems are considered. The existence of the variational eigenvalues is established. Asymptotic formulas of eigenvalues and eigenfunctions are obtained.(5) Asymptotic theory for solutions of elliptic systems : Semilinear elliptic systems are considered. We establish nonexistence criteria of positive solutions, Liouville-type theorems, and oscillation criteria.)