论文标题
涉及分数$ p $ laplacian和双重关键非线性的方程式系统
A system of equations involving the fractional $p$-Laplacian and doubly critical nonlinearities
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
This paper deals with existence of solutions to the following fractional $p$-Laplacian system of equations \begin{equation*} %\tag{$\mathcal P$}\label{MAT1} \begin{cases} (-Δ_p)^s u =|u|^{p^*_s-2}u+ \frac{γα}{p_s^*}|u|^{α-2}u|v|^β\;\;\text{in}\;Ω, (-Δ_p)^s v =|v|^{p^*_s-2}v+ \frac{γβ}{p_s^*}|v|^{β-2}v|u|^α\;\;\text{in}\;Ω, % % u,\;v\in\wsp, \end{cases} \end{equation*} where $s\in(0,1)$, $p\in(1,\infty)$ with $N>sp$, $α,\,β>1$ such that $α+β= p^*_s:=\frac{Np}{N-sp}$ and $Ω=\mathbb{R}^N$ or smooth bounded domains in $\mathbb{R}^N$. For $Ω=\mathbb{R}^N$ and $γ=1$, we show that any ground state solution of the above system has the form $(λU, τλV)$ for certain $τ>0$ and $U,\;V$ are two positive ground state solutions of $(-Δ_p)^s u =|u|^{p^*_s-2}u$ in $\mathbb{R}^N$. For all $γ>0$, we establish existence of a positive radial solution to the above system in balls. For $Ω=\mathbb{R}^N$, we also establish existence of positive radial solutions to the above system in various ranges of $γ$.
