论文标题
关键第六阶方程的积极奇异解决方案的分类
Classification for positive singular solutions to critical sixth order equations
论文作者
论文摘要
我们将整个积极的奇异解决方案分类为刺穿空间中关键的第六阶方程的家族，其起源不可移动。更确切地说，我们表明，当原点是不可易换的奇异性时，溶液是由奇异径向因子乘以恒定系数的第六阶IVP的定期溶液给出的。在技术层面上，我们将积分滑动方法和基于能源结果的质量分析结合在一起，以执行拓扑两参数射击技术。我们首先使用方程式的积分表示来运行移动球技术，该技术证明了解决方案相对于原点是径向对称的。因此，在Emden（福勒）坐标中，我们可以将问题减少到对具有恒定系数的第六阶自主颂歌的研究。我们论点背后的主要启发式方法是，由于ODE操作员的所有指示根都是正面的，因此可以将其分解为满足比较原理的三个秒运算符的组成。这使我们能够定义一个沿解决方案保守的哈密顿能量，我们从中提取了它们的定性特性，例如唯一性，有限性，渐近行为和分类。
We classify entire positive singular solutions to a family of critical sixth order equations in the punctured space with a non-removable singularity at the origin. More precisely, we show that when the origin is a non-removable singularity, solutions are given by a singular radial factor times a periodic solution to a sixth order IVP with constant coefficients. On the technical level, we combine integral sliding methods and qualitative analysis of ODEs, based on a conservation of energy result, to perform a topological two-parameter shooting technique. We first use the integral representation of our equation to run a moving spheres technique, which proves that solutions are radially symmetric with respect to the origin. Thus, in Emden--Fowler coordinates, we can reduce our problem to the study of an sixth order autonomous ODE with constant coefficients. The main heuristics behind our arguments is that since all the indicial roots of the ODE operator are positive, it can be decomposed into the composition of three second order operators satisfying a comparison principle. This allows us to define a Hamiltonian energy which is conserved along solutions, from which we extract their qualitative properties, such as uniqueness, boundedness, asymptotic behavior, and classification.
