Lane-Emden型方程组的临界指标

临界指标的研究是前沿热点课题。本项目研究椭圆型、抛物型、退化型这三类Lane-Emden方程组，来源天文学、物理学、生物数学等实际问题。第一部分研究椭圆型方程组不存在正解的临界指标，它与Lane-Emden猜测、刘维尔型定理、Sobolev指标等重要问题联系紧密。为了克服高维空间的技术困难，将在球坐标系作解的积分型估计、使用插值不等式与嵌入定理等方法，研究方案优点是空间降低一维，所得结果更优。第二与第三部分，研究抛物型与退化型方程组存在周期解的临界指标，这两类方程组作为数学模型能近似描述周期现象。方程组没有变分结构，难以用Pohozaev恒等式得到解的最大模估计，项目将采用改良的尺度变换、算子半群估计、拓扑度等方法，研究方案优点是去掉两类技术限制条件：凸区域与周期系数的结构条件。预期研究成果能得到解的临界指标、讨论方程组解的适定性与渐近行为，解释数学模型的结果与现实观测现象存在差异的原因。

The study of the critical exponent is a frontier hot topic. This project studies three types of Lane-Emden system, elliptic, parabolic, and degenerate types, which come from practical problems such as astronomy, physics and biological mathematics. The first part studies the nonexistence of positive solutions for the elliptic Lane-Emden system, which is closely related to important problems such as the Lane-Emden conjecture, the Liouville type theorem, and the Sobolev exponent, etc. In order to overcome the technical difficulties in high-dimensional space, the project will perform integral estimation for the solutions in the spherical coordinate system and use interpolation inequalities and embedding theorems, the advantage of this research schemes is that space is reduced by one dimension, and the results obtained are better. In the second and third parts, the project will study the critical exponent of the periodic solutions for parabolic and degenerate systems. These two types of systems can be used as mathematical models to approximately describe periodic phenomena. These systems have no variational structure and are difficult to obtain the maximum modulus estimation for solutions by the Pohozaev type identity. This project will use improved scaling methods, the estimation for operator semigroup, topological degree methods, etc. The advantage of this research scheme is to remove two types of technical restrictions: convex assumption on the domain and structural conditions of periodic coefficients. The research results are expected to obtain the critical exponent of solutions, the research results can be used to discuss the well-posedness of solutions for the systems and the asymptotic behavior of the solutions, explaining the reasons for the difference between the results of mathematical models and the phenomena observed in reality.