Entire large solutions to semilinear elliptic equations with rapidly or regularly varying nonlinearities

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 in Definitions 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 Δu=b(x)f(u),x∈RN(N≥3), where b∈C(RN) is positive in RN, 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∞.