当k>=2时，k-Hessian算子是完全非线性的，该算子有两个特殊的情形分别为k=1时的Laplace算子和k=n时的Monge-Ampère算子。对于这两类算子已有比较完善的研究结果，包括其所对应方程的解的存在性、唯一性、渐近性态以及稳定性。k-Hessian方程作为连接这两类算子对应方程的桥梁，在完全非线性偏微分方程理论研究中扮演着重要的角色，同时该方程解的性质在数学物理和微分几何学科中具有重要应用。 
本文应用先验估计、泰勒公式以及Hopf分岔理论等方法研究了一类带有Matukuma型源的k-Hessian方程径向正解的性质。研究表明该方程的径向正解可以分为两种不同的类型：E型解和M型解，它们分别为正则解和奇异解，其中奇异解包含正则部分和奇异部分。在研究E型解在原点附近的渐近行为时，本文利用泰勒展开式进行循环迭代逐步改进E型解在原点附近的渐近展开式的精度。由于该类型的解为正则解，每次迭代只会出现新的正则部分，即提高了其渐近展开式的精度。
基于上述研究所得到相关结果以及循环迭代的思想，本文研究了M型解在原点附近的渐近行为。在k-Hessian方程的研究历程中，对于M型解的研究是相对较少的，且与E型解相比，它具有更加丰富的性质。因此在M型解的研究过程中，本文将以p为主指标的区间分为四个子区间并分别利用不同的方法进行研究。对于该指标的超临界情况，首先将子区间再次分为三个小区间，其次建立先验估计，最后利用相应的积分微分方程和泰勒公式进行循环迭代逐步提高M型解渐近展开式的精度。在子区间的次临界情况中，与k=1的情况相比，本文多次使用了泰勒公式并给出了相关参数更精确的范围以保证提高M型解的渐近展开式的精度。由于该类型的解为奇异解，在研究过程中需要反复进行迭代直到有正则部分的出现，之后的处理方法与E型解类似。对于该指标的临界和次临界情况，此时上述的积分微分方程不再成立，本文利用渐近自治的Lotka-Volterra系统和相应的逆变换得到了M型解的渐近展开式。

The k-Hessian operator is completely nonlinear at k>=2 , which has two special cases: Laplace operator of k=1 and Monge-Ampere operator of k=n . The results of these two types of operators have been well developed, including the existence, uniqueness, asymptotic properties and stability of the solutions of the corresponding equations. As a bridge connecting the corresponding equations of these two types of operators, the k-Hessian equation plays an important role in the theoretical study of completely nonlinear partial differential equations, and the properties of the solution of the equation have important applications in mathematical physics and differential geometry.
In this paper, the properties of radial positive solutions of a class of k-Hessian equations with Matukuma-type sources are studied by using prior estimation, Taylor's formula and Hopf bifurcation theory. The results show that the radial positive solution of the equation can be divided into two different types: E-type solution and M-type solution, which are regular solution and singular solution, respectively, and the singular solution contains a regular part and a singular part. In order to study the asymptotic behavior of the E-type solution near the origin, this paper uses the Taylor expansion to perform cyclic iteration to gradually improve the accuracy of the Asymptotic expansion near the origin of the E-type solution. Since this type of solution is a regular solution, only a new regular part appears in each iteration, which improves the accuracy of its asymptotic expansion.
Based on the relevant results obtained from the above research and the idea of cyclic iteration, the asymptotic behavior of the M-type solution near the origin are studied. In the course of the study of the k-Hessian equation, the M-type solution is relatively rare, and it has more abundant properties than the E-type solution. Therefore, in the research process of M-type solution, this paper divides the interval of p as the main indicator into four subintervals and uses different methods to study them. For the supercritical case of the index, firstly, the subinterval is divided into three subintervals, then a priori estimation is established, and finally the accuracy of the asymptotic expansion of the M-type solution is gradually improved by using the corresponding integral differential equation and Taylor's formula. In the subcritical case of the subinterval, compared with the case of k=1, Taylor's formula is used several times and a more precise range of relevant parameters is given to ensure that the accuracy of the asymptotic expansion of the M-type solution is improved. Since this type of solution is a singular solution, it is necessary to iterate repeatedly in the research process until a regular part appears, and then the treatment method is similar to that of the E-type solution. For the critical and subcritical cases of this indicator, the above integral-differential equations no longer hold, and the asymptotic expansion of the M-type solution is obtained by using the asymptotic autonomous Lotka-Volterra system and the corresponding inverse transformation.

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