带对流项的非线性椭圆型问题爆炸解的存在性与渐近行为  

EXISTENCE AND ASYMPTOTIC BAHAVIOR OF EXPLOSIVE SOLUTIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH CONVECTION TERMS

作　　者：张志军[1] 

机构地区：[1]烟台大学数学与信息科学系

出　　处：《数学年刊（A辑）》2002年第3期395-406,共12页Chinese Annals of Mathematics

基　　金：国家自然科学基金(No.10071066)资助的项目.

摘　　要：设Ω是RN（N≥3）中的C2有界区域,对带负对流项的情形,对更广泛的非线性项,构造一种新型的非线性变换将爆炸解问题,转化成等价的带奇异项的Dirichlet问题,应用极大值原理得到了爆炸解问题解的最小爆炸速度.应用三种摄动方法,结合上下解方法、二阶椭圆型偏微分方程的估计理论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RN,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.而对带正对流项的情形,对更广泛的非线性项,构造爆炸上下解u和u在Ω上满足u≤u,得到了爆炸解u的存在性且在Ω上满足u≤u≤u.Let Ω be a bounded domain with C2 boundary Ω in RN(N ≥ 3), for the more general nonlinearity, the new change of variable transforms the problem of explosive solutions with a negative convection term into the equivalent Dirichlet problem. He exposes that the explosive solutions have the lowest speed. Then, by the perturbed methods, and sub-supersolutions method, the existence of explosive solutions is obtained. In addition, he allows the coefficient of nonlinearity to be not only suitable unbounded on Ω but also zero on large parts of Ω including Ω. He also showed that the problem has one entire solution and characterized the asymptotic behavior of the solution near ∞ when Ω = RN. For a positive convection term, he constructs explosive supersolution u and explosive subsolution u satisfying u ≤ u in Ω and obtain the existence of explosive solutions for the problem.

关 键 词：非线性椭圆型方程 对流项 爆炸解 存在性 最小爆炸速度 

分 类 号：O175[理学—数学]