Study on Positive Solutions of Nonlinear Elliptic Boundary Value Problems Arising in Population Dynamics

在这项研究中，我们研究了欧氏空间中光滑有界域上一类Logistic类型的非线性椭圆型边值问题正解的存在性和性态。我们建立了一个关于非线性边界条件下正解的分支和稳定性的定理。利用Lyapunov-Schmidt型程序、Crandall和Rabinowitz的局部分叉理论和上下解方法对系统进行了局部分叉分析。在第一个分支解的基础上，我们将问题归结为一个约束极小化问题，从而得到了正解的多重性。当参数足够小时，我们建立了一个非线性边界条件下正解的不存在性定理，其中使用了变分技巧和上下解。研究了一类带Robin型边界条件的线性不定本征值问题。利用变分方法证明了正主本征值的存在唯一性。我们还根据给定的特征值问题中包含的变号权，得到了正主特征值的先验下界。我们指出，如果将Logistic类型的非线性椭圆型边值问题线性化到等价于零的平凡解，就可以得到所考虑的线性特征值问题。研究了一类带Neumann边界条件的带不定权的线性椭圆型特征值问题的正主特征值的爆破性质。讨论了爆破性的充要条件。应该强调的是，我们构造了一个反例，以表明在Dirichlet边界条件情况下，已知的爆破性的充要条件不再成立。

In this research we studied existence and behavior of positive solutions to a nonlinear elliptic boundary value problem of logistic type in a smooth bounded domain of the Euclidean space, as a parameter included varies.1. We established a theorem on bifurcation and stability of positive solutions under nonlinear boundary conditions. A local bifurcation analysis was carried out by using the Lyapunov-Schmidt type of procedure, the local bifurcation theory due to Crandall and Rabinowitz and the method of super and subsolutions. Moreover we reduced our problem to a constrained minimization one based on the first bifurcation solution, to obtain multi plicity of positive solutions.2. We established a nonexistence theorem for positive solutions under nonlinear boundary conditions when parameter is small enough, where a variational technique and super and subsolutions are used.3. We studied a linear indefinite eigenvalue problem with a Robin type boundary condition. Existence and uniqueness of a positive principal eigenvalue was proved by use of a variational approach. We also obtained an a priori lower bound for the positive principal eigenvalue in terms of the sign-changing weight included in the given eigenvalue problem. We remark that we can get the linear eigenvalue problem under consideration if the nonlinear elliptic boundary value problem of logistic type is linearized at the trivial solution which is identically equal to zero.4. We studied blowing-up property of the positive principal eigenvalue for a linear elliptic eigenvalue problem with an indefinite weight and Neumann boundary condition. Necessary and sufficient conditions for the blowing-up property were discussed. It should be emphasized that we constructed a counterexample to show that a known necessary and sufficient condition for the blowing-up property in the Dirichlet boundary condition case no longer remains true.