Qualitative studies of solutions to elliptic equations in unbounded domains

无界域中椭圆方程解的定性研究

• 批准号: 09640192
• 负责人: USAMI Hiroyuki
• 金额: $ 1.02万
• 依托单位: Hiroshima University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640192/
• 关键词: elliptic equation; degenerate Laplacian; Liouville-type theorem; Sturm-Liouville problem; oscillation; eigenvalue problem; symmetry of solutions; elliptic system; 非線形楕円型方程式; リューヴィユ型定理; 非線形固有値問題; elliptic equation; degenerate Laplacian; Liouville-type theorem; Sturm-Liouville problem; oscillation; eigenvalue problem; symmetry of solutions; elliptic system; 非線形楕円型方程式; リューヴィユ型定理; 非線形固有値問題

(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.)

（1）椭圆方程的振荡标准：为二阶准线性椭圆方程式建立了有效的振荡标准，其主要术语是退化的laplacians。我们的方法基于准线性分化方程的比较原理和渐近理论。对于一维情况，通过广泛的普劳转换获得了有关解决方案零的有用信息。（2）liouville-type定理和BVPS阳性解决方案的不存在：lioville-type定理的阳性解决方案是对准椭圆形方程式确定的，其主要椭圆形方程的主要术语是变性的，并且具有普通的laplac和普通级别（普遍）。我们的结果可以被视为经典liouville定理的自然扩展。（3）椭圆形问题的阳性溶液的对称性：我们通过移动平面方法或移动球方法证明某些类型的椭圆方程的阳性溶液是辐射对称的。有关抛物线问题的自相似解决方案的有用信息可以从我们的结果中得出。（4）两参数特征值问题：两参数非线性sturm sturm-liouville问题。建立了变异特征值的存在。获得了本征值和本征函数的渐近公式。（5）考虑到椭圆系统溶液的渐近理论：半线性椭圆系统。我们建立了积极解决方案，liouville型定理和振荡标准的不存在标准。）