|Dec 22, 2021
We are concerned with positive solutions of the Lane-Emden system on a smooth bounded convex domain. This system appears as an extremal equation of a particular Sobolev embedding and is also closely related to the Calderon-Zygmund estimate. Given an arbitrary family of solutions, we thoroughly analyze its asymptotic behavior as the exponents of the nonlinearities tend to the critical ones, establishing a detailed qualitative and quantitative description. In particular, we derive a priori energy bound for the solutions and prove that the multiple bubbling phenomena may arise. Also, we observe a dichotomy phenomenon for the asymptotic behavior of the family of solutions. In one case, the nonlinear structure of the system makes the interaction between bubbles so strong, so the determination process of the blow-up rates and locations is totally different from that of the classical Lane-Emden equation. In the other case, the blow-up scenario is relatively close to (but not the same as) that of the classical Lane-Emden equation, and only one-bubble solutions can exist. Even in the latter case, the standard approach does not work well, which forces us to devise a new method. As a by-product of our analysis, we also obtain a general existence theorem valid on any smooth bounded domains. This is joint work with Sang-Hyuck Moon (National Center for Theoretical Sciences, Taiwan).