半线性椭圆型问题爆炸解的存在性与渐近行为

设Ω是RN（N≥3）中的C2有界区域,f是单调非减的非负连续可微函数满足f＇（a）∫a∞1/f（s）ds≤C0,　a>0.应用一种新型的非线性变换w（x）=∫u（x）∞　ds/f（s）将爆炸解问题△u=k（x）f（u）,u>0,x∈Ω,u|　Ω=∞转化成等价的带奇异项的Dirichlet问题,不仅得到了爆炸解问题解的最小爆炸速度,而且揭示了两类典型非线性爆炸解问题基本上是相同的.应用摄动方法,上下解方法得到了爆炸解的存在性.特别允许非线性项的系数不仅在Ω的内部子区域恒为零而且在Ω上可适当无界.随后再应用摄动方法,将所得结果推广到无界区域,得到了整体爆炸解的存在性以及在无穷远附近的最小爆炸速度（有关文献参见1-33]）.

The Existence and Asymptotic Bahavior of Explosive Solutions for Semilinear Elliptic Problems

They consider the semilinear problems △u=k(x)f(u), u>0, x∈Ω, u| Ω=∞, where Ω a bounded domain with C2 boundary Ω in RN (N≥3), f is a nonneg-ative, nondecreasing C1 function satisfying f'(a)∫a∞1/f(s)ds≤C0, a > 0. The new change of variable w(x) =∫u(x)∞ds/f(s) transforms the problems of explosive solutions into the equivalent singular Dirichlet problems. They expose that the explosive solutionshave the lowest speed and the two models of explosive problems are basically the one. Then, by the perturbed method, and sub - supersolutions method, the existence of explosive solutions is obtained. In addition, They allow k to be not only suitable unbounded on Ω but also zero on large parts of Ω including Ω. They also show that the problems have one entire solution and characterize the asymptotic behavior of the solution near ∞ when Ω= RN(see 1-33]).