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

Abstract

We concern a family { ( u ε , v ε ) } ε > 0 \{(u_{\varepsilon },v_{\varepsilon })\}_{\varepsilon > 0} of solutions of the Lane-Emden system on a smooth bounded convex domain Ω \Omega in R N \mathbb {R}^N \[ { − Δ u ε = v ε p a m p ; in Ω , − Δ v ε = u ε q ε a m p ; in Ω , u ε , v ε > 0 a m p ; in Ω , u ε = v ε = 0 a m p ; on ∂ Ω , \begin {cases} -\Delta u_{\varepsilon } = v_{\varepsilon }^p & \text {in } \Omega , \\ -\Delta v_{\varepsilon } = u_{\varepsilon }^{q_{\varepsilon }} & \text {in } \Omega , \\ u_{\varepsilon },\, v_{\varepsilon } > 0 & \text {in } \Omega , \\ u_{\varepsilon } = v_{\varepsilon } =0 & \text {on } \partial \Omega , \end {cases} \] for N ≥ 4 N \ge 4 , max { 1 , 3 N − 2 } > p > q ε \max \{1,\frac {3}{N-2}\} > p > q_{\varepsilon } and small \[ ε ≔ N p + 1 + N q ε + 1 − ( N − 2 ) > 0. \varepsilon ≔\frac {N}{p+1} + \frac {N}{q_{\varepsilon }+1} - (N-2) > 0. \] This system appears as the extremal equation of the Sobolev embedding W 2 , ( p + 1 ) / p ( Ω ) ↪ L q ε + 1 ( Ω ) W^{2,(p+1)/p}(\Omega ) \hookrightarrow L^{q_{\varepsilon }+1}(\Omega ) , and is also closely related to the Calderón-Zygmund estimate. Under the natural energy condition, we prove that the multiple bubbling phenomena may arise for the family { ( u ε , v ε ) } ε > 0 \{(u_{\varepsilon },v_{\varepsilon })\}_{\varepsilon > 0} , and establish a detailed qualitative and quantitative description. If p > N N − 2 p > \frac {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 p \ge \frac {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 p near N N − 2 \frac {N}{N-2} . We also deduce a general existence theorem that holds on any smooth bounded domains.