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

考虑下面的半线性退化偏微分方程：-g（|x|）Δu + u|u|^p - f(|x|) = 0∈R^N(1)其中g是一个阶为ell |的非负多项式，在无穷远处的一个点的邻域内f也是一个无穷远处的一个点的邻域内的多项式。此外，假设g的零点有界。因此，微分方程是简并型的。我们不对(1)的解在无穷近处施加边界条件。因此，有可能存在多个连续的粘度解。我们的研究目的是分析(1)的许多连续粘度解的集合的结构。我们的研究结果如下。首先证明了N与k的关系的不等式是决定根对称解是无穷还是单的充分必要条件，其中k是f的一个最大阶系数，并且证明了许多根对称解的集合同胚于R^1。其次，在N = 2的假设条件下，在f， g的多项式的低阶项不存在的情况下，找到了判断是否存在非根本对称解的条件。如果非根本对称解存在我们也展示了有多少非根本对称解。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.