含梯度项椭圆边值问题正径向解的存在性

Existence of Positive Radial Solutions for EllipticBoundary Value Problems with Gradient Terms

用上下解方法讨论球外部区域Ω={x∈ℝ^(N):x>R 0}上含梯度项的椭圆边值问题:{-Δu=K(|x|)f(|x|,u,u),x∈Ω,αu+β∂u/∂n|∂Ω=0,lim|x|→∞u(x)=0正径向解的存在性与唯一性,其中N≥3,R 0>0,K:[R 0,∞)→ℝ+和f:[R 0,∞)×ℝ×ℝ+→ℝ连续.在系数函数K(r)=O(1/r 2(N-1))(r→+∞),非线性项f(r,u,η)满足一些适当的不等式条件且关于η满足Nagumo条件时,证明该问题正径向解的存在性与唯一性.

Using the method of upper and lower solutions,we discuss the existence and un iqueness of positive radial solutions for elliptic boundary value problems with gradient term:-Δu=K(|x|)f(|x|,u,u),x∈Ω,αu+β∂u∂/n∂Ω=0,lim x→∞u(x)=0,whereΩ={x∈ℝN:x>R 0},N≥3,R 0>0,K:[R 0,∞)→ℝ+and f:[R 0,∞)×ℝ×ℝ+→ℝare continu ous.When the coefficient function K(r)=O(1/r^2(N-1))(r→+∞),under the condition that the nonlinear term f(r,u,η)satisfies some approp riate inequality conditions and Nagumo condition onη,we prove the existence and uniqueness of the positive radial solution of the problem.