Uniqueness of boundary blow-up solutions on unbounded domain of R(N)

作者：Dong Wei; Li Juanfei; Liu Lishan

来源：Nonlinear Analysis-Theory Methods & Applications, 2009, 71(12): E2118-E2126.

DOI：10.1016/j.na.2009.03.084

• The first eigenvalue
• Sub-super solution
• Comparison principle
• Positive solution
• Boundary blow-up solution

In this paper, we consider the existence of positive solutions of the degenerate logistic type elliptic equation
-Delta u = a(x)u - b(x)f(u), x is an element of R(N)\ Du vertical bar(partial derivative D) = infinity,
where N >= 2, D subset of R(N) is a bounded smooth domain and a(x), b(x) are continuous functions on R(N) with b(x) >= 0, b(x) not equivalent to 0, especially b(x) = 0 on partial derivative D and f(u) is an element of C(0, infinity). We show that under rather general conditions on a(x) and b(x) for large vertical bar x vertical bar and f(u) behaves like u(q) where q > 1. Without the behavior of b(x) near the boundary partial derivative D, it will be shown that there exists a unique positive solution. Our results improve the corresponding ones in [W. Dong, Pang, Uniqueness of boundary blow-up solutions on exterior domain of R(N), J. Math. Anal. Appl. 330 (2007), 654-664] and [Y. Du,L. Ma, Logistic type equations on R(N) by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., (2) 64 (2001), 107-124].

• 出版日期2009-12-15
• 单位河北大学; 河北工程大学; 曲阜师范大学