Asymptotic analysis on positive solutions of the Lane-Emden system with nearly critical exponents

Abstract

We concern a family {(u(epsilon), v(epsilon))}(epsilon)>0 of solutions of the Lane-Emden system on a smooth bounded convex domain Omega in R-N [GRAPHICS] for N >= 4, max{1, 3/N-2} < p < q(epsilon) and small [GRAPHICS] This system appears as the extremal equation of the Sobolev embedding W-2,W-(p+1)/p(Omega) -> Lq epsilon+1(omega), and is also closely related to the Calderon-Zygmund estimate. Under the natural energy condition, we prove that the multiple bubbling phenomena may arise for the family {(u(epsilon), v(epsilon))}(epsilon)>0, and establish a detailed qualitative and quantitative description. If p < N/N-2, 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 completely different from that of the classical Lane-Emden equation. If p >= N/N-2, the blow-up scenario is relatively close to that of the classical Lane-Emden equation, and only single-bubble solutions can exist. Even in the latter case, we have to devise a new method to cover all p near N/N-2. We also deduce a general existence theorem that holds on any smooth bounded domains.