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

考虑一类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是无穷远点邻域内的1&gt；2次非负多项式，f也是无穷远点邻域内的多项式.此外，假设g有有界零点。因此，该微分方程是一种退化形式。我们不将边界条件强加给(1)在点无穷远附近的解。我们的研究目的是分析(1)的多个连续粘性解的集合的结构。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径向对称解是无穷的还是唯一的充要条件，其中k是f的极大阶系数.此外，我们还证明了一组多个径向对称解是R^1的同胚.其次，在假设N=2且f，g的多项式的低阶项不存在的情况下，我们找到了判定是否存在非径向对称解的条件.如果存在非径向对称解，我们还证明了存在多少个非径向对称解。也就是说，我们证明了解的数目取决于与L，p，k相关的值的不同。

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.