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

研究了欧氏空间中光滑有界区域上一类非线性Logistic型椭圆边值问题，当参数包含变量时，正解的存在性和性态.建立了非线性边界条件下正解的分支和稳定性定理。利用Lyapunov-Schmidt型过程、Crandall和Rabinowitz的局部分歧理论和上下解方法对系统进行了局部分歧分析。此外，基于第一个分歧解，我们将问题转化为一个约束极小化问题，得到了多个正解.利用变分技巧和上下解的方法，在参数足够小的情况下，建立了非线性边界条件下正解的不存在性定理.研究了一类具有Robin型边界条件的线性不定特征值问题.利用变分方法证明了正主特征值的存在唯一性。我们还得到了一个先验下界的正的主特征值的符号变化的重量包括在给定的特征值问题。我们注意到，如果在恒等于零的平凡解处将非线性Logistic型椭圆边值问题线性化，则我们可以得到所考虑的线性特征值问题.研究了具有不定权和Neumann边界条件的线性椭圆特征值问题正主特征值的爆破性质。讨论了爆破的充分必要条件。应该强调的是，我们构造了一个反例来证明Dirichlet边界条件下爆破性质的一个已知的充分必要条件不再为真。

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.