论文标题
具有逆方势的车道填充系统的正效应
Positive supersolutions for the Lane-Emden system with inverse-square potentials
论文作者
论文摘要
在本文中，我们研究了以下具有逆向电位的车道填充系统的阳性超溶液的不存在 \ begin {qore} \ label {0} \ left \ {\ begin {array} {lll} -ΔU+\ frac {μ_1} {| x |^2} u = v^p \ quad {\ rm in} \ \，ω\ setMinus \ {0 \}，\\ [2mm] -ΔV+\ frac {μ_2} {| x |^2} v = u^q \ quad {\ rm in} \ \，ω\ setMinus \ {0 \} \ end {array} \正确的。 \ end {equation}对于合适的$ p，q> 0 $，$μ_1，μ_2\ geq-（n -2）^2/4 $，其中$ω$是一个平稳的有限域，包含$ \ m athbb {r}^n $中的原点，带有$ n \ geq 3 $。确切地说，在$ - （N-2）^2/4 \ leqμ_1，μ_2<0 $和$ - （N-2）^2/（N-2）^2/4 \ leq leq leq leq leq uqμ_1<0 \ 0 \ leq leq leq leq leq leq leq leq leq leq leq leq leq leq leq $ q $ q $ q q的情况下，我们为系统（\ ref {0}）提供了较高的$（p，q）$（p，q）$，以实现系统（\ ref {0}）的不存在。由于负系数为$μ_1，反向平方电位的μ_2$可以得出原点的初始爆炸，并且可以在超临界情况下应用迭代程序，以提高爆炸率，直到在某些加权$ l^1 $ l^1 $中不可允许的非线性。在亚临界情况下，我们通过特定的径向对称函数证明了系统（\ ref {s 1.1}）的阳性超溶液的存在。
In this paper, we study the nonexistence of positive supersolutions for the following Lane-Emden system with inverse-square potentials \begin{equation}\label{0} \left\{ \begin{array}{lll} -Δu+\frac{μ_1}{|x|^2} u= v^p \quad {\rm in}\ \, Ω\setminus\{0\},\\[2mm] -Δv+\frac{μ_2}{|x|^2} v= u^q \quad {\rm in}\ \, Ω\setminus\{0\} \end{array} \right. \end{equation} for suitable $p,q>0$, $μ_1,μ_2\geq -(N-2)^2/4$, where $Ω$ is a smooth bounded domain containing the origin in $\mathbb{R}^N$ with $N\geq 3$. Precisely, we provide sharp supercritical regions of $(p,q)$ for the nonexistence of positive supersolutions to system (\ref{0}) in the cases $-(N-2)^2/4\leq μ_1,μ_2<0$ and $-(N-2)^2/4\leq μ_1<0\leq μ_2$. Due to the negative coefficients $μ_1,μ_2$ of the inverse-square potentials, an initial blowing-up at the origin could be derived and an iteration procedure could be applied in the supercritical case to improve the blowing-up rate until the nonlinearities are not admissible in some weighted $L^1$ spaces. In the subcritical case, we prove the existence of positive supersolutions for system (\ref{s 1.1}) by specific radially symmetric functions.
