Research of the structure of unbounded viscosity solutions to semilinear degenerate elliptic equations in R^N

考虑如下半线性退化偏微分方程：-g（|X|）Δu + u| u| 10 - 12 - 2000（|X|）= 0 ∈ R^N（1）其中g是在无穷远点邻域内的次e 11> 2的非负负范数，f也是在无穷远点邻域内的非负范数.此外，假设g保持有界零点。因此，该微分方程是退化型的。在无穷远点的邻域内，我们不对方程（1）的解施加边界条件。本文的主要目的是分析方程（1）的多个连续粘性解的结构。我们的研究结果如下。我们首先证明了N与k的关系的一个不等式是判定根对称解是无穷大还是单解的一个充分必要条件，其中k是f的一个极大阶系数。此外，我们还证明了一组多个根对称解与R^1同胚。其次，在N = 2和f，g的多项式的低阶项不存在的条件下，给出了判断方程是否存在非根对称解的条件。如果存在非根对称解，我们也给出了有多少个非根对称解。也就是说，我们证明了解的数量是不同的，取决于与l，p，k相关的值。

Consider a following semilinear degenerate partial differential equation:-g(|x|)Δu + u|u|^p - f(|x|) = 0 ∈ R^N (1)where g is a nonnegative plynominal of degree ell > 2 in a neighborhood of a point at infinity and f is also a plynominal in a neighborhood of a point at infinity. Moreover, assume g holds bounded zero points. Hence, the differential equation is a degenerate tyle. We don't impose the boundary condition to solutions of (1) in the neighborhood of the point infinity. Then, It is possible to exist many continuous viscosity solutions.Our purpose of this research was to analyze the structure of the set of many continuous viscosity solutions of (1). The results of our research are as follows. We the first showed that an inequality of relations of N and k is a necessary and sufficient condition to decide whether radically symmetric solutions are infinite or single where k is a coefficient of a maxima order of f. Moreover, we proved that a set of many radically symmetric solutions is homeomorphic to R^1. The secondly, under the assumptions N = 2 and that lower order terms of polynominal of f, g do not exist, we found the condition to judge whether there exists non radically symmetric solution or not. If non-radically symmetric solution exists we also showed how many non-radically symmetric solutions there were. That is, We showed that the number of solutions was different depending on the value that related to l, p, k.