arXiv 2402.090792024-02math.AP

涉及 $p$-Laplacian $-\Delta_{p}$ 的 $D^{1,p}$-临界拟线性静态薛定谔-哈特里方程非负解的径向对称性与尖锐渐近行为

Radial symmetry and sharp asymptotic behaviors of nonnegative solutions to $D^{1,p}$-critical quasi-linear static Schrödinger-Hartree equation involving $p$-Laplacian $-Δ_{p}$

Wei Dai, Yafei Li, Zhao Liu

备注：: Extended results on radial symmetry and sharp asymptotic estimates for more general nonlocal quasi-linear equations were added in this version

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AI总结：

本文研究含 $p$-Laplacian 的 $D^{1,p}$-临界拟线性静态薛定谔-哈特里方程，通过 De Giorgi-Moser-Nash 迭代和移动平面法，证明非负解的正则性、渐近行为尖锐估计及径向对称性。

摘要：

本文主要考虑具有 $p$-Laplacian $-\Delta_{p}$ 和非局部非线性的 $D^{1,p}(\mathbb{R}^{N})$-临界拟线性静态薛定谔-哈特里方程的非负弱解：\begin{align*} -\Delta_p u =\left(|x|^{-2p}\ast |u|^{p}\right)|u|^{p-2}u \qquad &\mbox{in} \\, \mathbb{R}^N, \end{align*} 其中 $1<p<\frac{N}{2}$，$N\geq3$，且 $u\in D^{1,p}(\mathbb{R}^N)$。与文献 \cite{CFR,LDSMLMSB,GV,Ou,BS16,VJ16} 等研究的 $D^{1,p}(\mathbb{R}^{N})$-临界局部非线性项 $u^{p^{\star}-1}$（其中 $p^{\star}:=\frac{Np}{N-p}$）不同，由于哈特里型非线性项中出现了非局部卷积 $|x|^{-2p}*u^p$，我们无法像 \cite{VJ16} 中那样使用缩放论证和加倍引理来获得任何正解 $u$ 的渐近行为上界的初步估计。此外，要获得拟范数 $\\|u \\|_{L^{s,\infty}(\mathbb{R}^N)}$ 的有界性，并由此从初步估计推导出渐近行为上界的尖锐估计（如 \cite{VJ16} 中那样），也是相当困难的。幸运的是，通过 De Giorgi-Moser-Nash 迭代方法证明了渐近行为上界更好的初步估计，并结合 \cite{XCL} 的结果，我们能够克服这些困难，为更一般的方程 $-\Delta_p u=V(x)u^{p-1}$（其中 $V\in L^{\frac{N}{p}}(\mathbb{R}^{N})$）的任何正解 $u$ 建立正则性以及渐近行为上界和下界的尖锐估计。然后，利用 \cite{BS16,VJ16} 中的论证，我们可以推导出 $|\nabla u|$ 衰减率的上界和下界的尖锐估计。最后，作为推论，我们可以应用移动平面法证明所有非平凡非负解关于某点 $x_0\in\mathbb{R}^N$ 是径向对称且严格递减的。

英文摘要：

In this paper, we mainly consider nonnegative weak solution to the $D^{1,p}(\R^{N})$-critical quasi-linear static Schrödinger-Hartree equation with $p$-Laplacian $-Δ_{p}$ and nonlocal nonlinearity: \begin{align*} -Δ_p u =\left(|x|^{-2p}\ast |u|^{p}\right)|u|^{p-2}u \qquad &\mbox{in} \,\, \mathbb{R}^N, \end{align*} where $1<p<\frac{N}{2}$, $N\geq3$ and $u\in D^{1,p}(\R^N)$. Being different to the $D^{1,p}(\R^{N})$-critical local nonlinear term $u^{p^{\star}-1}$ with $p^{\star}:=\frac{Np}{N-p}$ investigated in \cite{CFR,LDSMLMSB,GV,Ou,BS16,VJ16} etc., since the nonlocal convolution $|x|^{-2p}*u^p$ appears in the Hartree type nonlinearity, it is impossible for us to use the scaling arguments and the Doubling Lemma as in \cite{VJ16} to get preliminary estimates on upper bounds of asymptotic behaviors for any positive solutions $u$. Moreover, it is also quite difficult to obtain the boundedness of the quasi-norm $\|u \|_{L^{s,\infty}(\R^N)}$ and hence derive the sharp estimates on upper bounds of asymptotic behaviors from the preliminary estimates as in \cite{VJ16}. Fortunately, by showing a better preliminary estimates on upper bounds of asymptotic behaviors through the De Giorgi-Moser-Nash iteration method and combining the result from \cite{XCL}, we are able to overcome these difficulties and establish regularity and the sharp estimates on both upper and lower bounds of asymptotic behaviors for any positive solution $u$ to more general equation $-Δ_p u=V(x)u^{p-1}$ with $V\in L^{\frac{N}{p}}(\mathbb{R}^{N})$. Then, by using the arguments from \cite{BS16,VJ16}, we can deduce the sharp estimates on both upper and lower bounds for the decay rate of $|\nabla u|$. Finally, as a consequence, we can apply the method of moving planes to prove that all the nontrivial nonnegative solutions are radially symmetric and strictly decreasing about some point $x_0\in\R^N$.

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