Studies on Eigenvalue Problems of Nonlinear Elliptic Equations

(1)椭圆型方程的特征值问题：研究半线性椭圆型方程的双参数特征值问题。我们建立（变分）特征值和特征函数的渐近性质。研究了两参数Ambrosetti-Prodi问题。我们调查参数和解决方案的数量之间的关系。(2)椭圆型方程的正解：在无界区域上考虑半线性二阶椭圆型方程。我们建立了正解的多重性结果和正解的唯一性定理。(3)拟线性常微分方程的正解：考虑首项为一维伪拉普拉斯算子的拟线性常微分方程。得到了正解的渐近表示。作为这些结果的应用，我们证明了拟线性椭圆型方程Dirichlet外问题几类正解的存在性。(4)描述霉菌聚集现象的数学模型：我们考虑由Keller和Segel引入的抛物系统的自相似解来描述由于趋化性导致的霉菌聚集现象。阐明了参数与自相似解个数之间的关系。(5)拟线性椭圆方程和椭圆方程组的非负非平凡解：我们建立了拟线性椭圆方程和拟线性椭圆方程组具有非平凡非负整解的必要和/或充分条件。还得到了几个Liouville型定理。

(1) Eigenvalue Problems of Elliptic Equations : Two-parameter eigenvalue problems for semilinear elliptic equations are studied. We establish asymptotic properties of (variational) eigenvalues and eigenfunctions. Two-parameter Ambrosetti-Prodi problems are also studied. We investigate the relation between parameters and the number of solutions.(2) Positive Solutions of Elliptic Equations : Semilinear second-order elliptic euations are considered in unbounded domains. We establish multiplicity results for positive solutions and uniqueness theorems for positive solutions.(3) Positive Solutions of Quasilinear Ordinary Differential Equations : Quasilinear ordinary differential equations whose leading term is one-dimensionai pseudo-Laplacian are considered. We obtain asynrptotic representations of positive solutions. As an application of these results, we show existence of several types of positive solutions of exterior Dirichlet problems for quasilinear elliptic equations.(4) Mathematical Models Describing Aggregation Phenomena of Molds : We consider self-similar solutions of parabolic systems introduced by Keller and Segel to describe aggregation phenomena of molds due to chemotaxis. We clarify the relation between parameters and the number of self-similar solutions.(5) Nonnegative Nontrivial Solutions of Quasilinear Elliptic Equations and Elliptic Systems : We establish necessary and/or sufficient conditions for quasilinear elliptic equations, as well as quasilinear elliptic systems, to possess nontrivial nonnegative entire solutions. Several Liouville type theorems are also obtained.