Entire large solutions to semilinear elliptic equations with rapidly or regularly varying nonlinearities. (February 2019) - Wan, Haitao; Li, Xiaohong; Li, Bo; Shi, Yongxiu
Abstract: In this paper, we first show some necessary and sufficient conditions to the rapidly and regularly varying functions defined in Appendix A; in particular, we find that the rapidly varying function classes ℜ ∞ and G ∞ given inDefinitions A.4 and A.5, respectively, are not equivalent, i.e., ℜ ∞ ⫋ G ∞ . Thereafter, we establish the asymptotic behavior and uniqueness of entire large solutions to the semilinear elliptic equation Δ u = b ( x ) f ( u ), x ∈ R N ( N ≥ 3 ), where b ∈ C ( R N ) is positive in R N, f ∈ C [ 0, ∞ ) is positive and non-decreasing on ( 0, ∞ ), and belongs to a more general class of functions. Additionally, we also study the asymptotic behavior and uniqueness of entire large solutions when f ∈ G ∞ ∖ ℜ ∞ .