论文标题
对带有几乎关键指数的车道填充系统的阳性解决方案的渐近分析
Asymptotic analysis on positive solutions of the Lane-Emden system with nearly critical exponents
论文作者
论文摘要
我们关注一个家庭$ \ {（u _ {\ varepsilon}，v _ {\ varepsilon}）\} _ {\ varepsilon> 0} $ emen-emden系统的解决方案的$ \ \ \ m naterbbbb { -ΔU_ {\ varepsilon} = v _ {\ varepsilon}^p＆\ text {in}ω，\\ - \ \ - \\ - ΔV_ {\ varepsilon} = u _ {\ varepsilon} u _ {\ varepsilon}，\，v _ {\ varepsilon}> 0＆\ text {in}ω，\\ \ u _ {\ varepsilon} = v _ {\ varepsilon} $ \ max \ {1，\ frac {3} {n-2} <p <p <q _ {\ varepsilon} $和小\ [\ varepsilon：= \ frac {n} {n} {p+1}系统显示为嵌入$ w^{2，（p+1）/p}（ω）\ hookrightArrow l^{q _ {\ varepsilon} +1}（ω）$的极端方程式，并且与calderón-Zygmund estimate也密切相关。在自然能量条件下\ | v _ {\ varepsilon} \ | _ {w^{2，{q _ {q _ {\ varepsilon} +1 \ over q _ {\ varepsilon}}}}}}}}}}}}}（ω）} \ right） $ \ {（u _ {\ varepsilon}，v _ {\ varepsilon}）\} _ {\ varepsilon> 0} $，并建立一个详细的定性和定量描述。如果$ p <\ frac {n} {n-2} $，则系统的非线性结构使气泡之间的相互作用如此强大，因此爆破率和位置的确定过程与经典的车道填充方程完全不同。如果$ p \ ge \ frac {n} {n-2} $，则爆炸场景相对接近（但与经典车道填充方程的（但不相同），并且只有一个泡沫解决方案可以存在。即使在后一种情况下，标准方法也无法正常工作，这迫使我们设计了一种新方法。使用我们的分析，我们还推断出对任何平滑有限域中有效的一般存在定理。
We concern a family $\{(u_{\varepsilon},v_{\varepsilon})\}_{\varepsilon > 0}$ of solutions of the Lane-Emden system on a smooth bounded convex domain $Ω$ in $\mathbb{R}^N$ \[\begin{cases} -Δu_{\varepsilon} = v_{\varepsilon}^p &\text{in } Ω,\\ -Δv_{\varepsilon} = u_{\varepsilon}^{q_{\varepsilon}} &\text{in } Ω,\\ u_{\varepsilon},\, v_{\varepsilon} > 0 &\text{in } Ω,\\ u_{\varepsilon} = v_{\varepsilon} =0 &\text{on } \partialΩ\end{cases}\] for $N \ge 4$, $\max\{1,\frac{3}{N-2}\} < p < q_{\varepsilon}$ and small \[\varepsilon := \frac{N}{p+1} + \frac{N}{q_{\varepsilon}+1} - (N-2) > 0.\] This system appears as the extremal equation of the Sobolev embedding $W^{2,(p+1)/p}(Ω) \hookrightarrow L^{q_{\varepsilon}+1}(Ω)$, and is also closely related to the Calderón-Zygmund estimate. Under the a natural energy condition \[\sup_{\varepsilon > 0} \left(\|u_{\varepsilon}\|_{W^{2,{p+1 \over p}}(Ω)} + \|v_{\varepsilon}\|_{W^{2,{q_{\varepsilon}+1 \over q_{\varepsilon}}}(Ω)}\right) < \infty,\] we prove that the multiple bubbling phenomena may arise for the family $\{(u_{\varepsilon},v_{\varepsilon})\}_{\varepsilon > 0}$, and establish a detailed qualitative and quantitative description. If $p < \frac{N}{N-2}$, 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 $p \ge \frac{N}{N-2}$, 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. Using our analysis, we also deduce a general existence theorem valid on any smooth bounded domains.
