On symmetric and radial viscosity solutions for elliptic partial differential equation.

点击翻译按钮获取中文摘要

We consider the Dirichelet problem for a semilinear degenerte elliptic equation (DP) : -g(|x|)Δu+f(|x|, u(x))=0, and Boundary Condition where N【greater than or equal】2 and g (|x|), f(|x|, u) are continuous and the domain is a bounded ball in N-dimensional space. We discuss the problem (DP) under the following assumptions : 1)g is nonnegative. 2) f is strictly monotone for u. We frist define a standard viscosity solution by the viscosity solution such that f (|x|, u(x))=0 if g(|x|)=0. Then we can prove that the any continuous standard viscosity solution is the radial solution and unique. We add an assumption : 3)∫^<a-0>g^<-1> (s) ds=∞ or ∫_<a+0> g^<-1> (s) ds=∞ for any a : g (a)=0. Then We obtain that any continuous viscosity solution is the radial solution and uniqne. If the assumption 3) is not satisfied there exist examples such that the continuous viscosity solutions are not uniqne.We next state the existence and uniqueness of the continuous unbounded viscosity solution in R^N. We u … More se the order of the infinite neiborhood of the solution as the boundary condition. We know that the existence or nonexistece of the solution are dependent on a kind of the order of the solution. Moreover, we get the results which the uniqueness or non-uniqueness are also dependent on a kind of the order of the solution. In this case, we assume that g, f is sufficiently smooth.We now show the existence and uniquness of the continuous viscosity solution to quasi-semilinear degenrate elliptic problem. Here, g (|x|, u), f (|x|, u) are continuous and f is strictly monotone for u. Moreover, we assume there exists an implicite function of f=0 and the implicite function holds some smootheness. Then we can prove the existence of the continuous viscosity solution. We next state the uniquenss of the continuous viscosity solution. Assume that g (|x|, u) and f (|x|, u) hold the some relations such that f (|x|, u)/g (|x|, u) is monotone for u. Then we have the uniquness theorem and get the result this viscosity solution is the radial solution. Less