标题： 非径向解对于涉及$p$-拉普拉斯算子的临界拟线性赫农方程在$\R^N$中 显示英文标题

标题： Non-radial solutions for the critical quasi-linear Hénon equation involving $p$-Laplacian in $\R^N$

摘要： 在本文中，我们研究以下包含$p$-拉普拉斯算子\begin{equation*}\label{00} \left\{ \begin{aligned} &-\Delta_p u=|x|^{\alpha}u^{p_\al^*-1}, & x\in \R^N, \\ &u>0, & x\in \R^N, \end{aligned} \right. \end{equation*}的$D^{1,p}$-临界拟线性 Hénon 方程，其中$N\geq2$，$1<p<N$，$p_\al^*:=\frac{p(N+\al)}{N-p}$和$\alpha>0$。 通过仔细研究线性化问题并应用逼近方法和分岔理论，我们证明当参数$\al$取临界值$\al(k):=\frac{p\sqrt{(N+p-2)^2+4(k-1)(p-1)(k+N-1)}-p(N+p-2)}{2(p-1)}$对于$k\geq2$时，上述拟线性 Hénon 方程存在非径向解$u$，使得$u\sim |x|^{-\frac{N-p}{p-1}}$和$|\nabla u|\sim |x|^{-\frac{N-1}{p-1}}$在$\infty$处。 应注意，当$p=2$时，$\alpha(k)=2(k-1)$对$k\geq2$成立。 我们的结果成功地扩展了 F. Gladiali, M. Grossi 和 S. L. N. Neves 在\cite{GGN}中关于拉普拉斯算子（即情况$p=2$）的经典工作，到非线性$p$-拉普拉斯算子（$1<p<N$）更一般的设置。 我们克服了一系列关键困难，包括$p$-Laplacian$\Delta_p$的非线性特性，缺乏 Kelvin 类型变换以及缺乏 Green 积分表示公式。 显示英文摘要

摘要： In this paper, we investigate the following $D^{1,p}$-critical quasi-linear H\'enon equation involving $p$-Laplacian \begin{equation*}\label{00} \left\{ \begin{aligned} &-\Delta_p u=|x|^{\alpha}u^{p_\al^*-1}, & x\in \R^N, \\ &u>0, & x\in \R^N, \end{aligned} \right. \end{equation*} where $N\geq2$, $1<p<N$, $p_\al^*:=\frac{p(N+\al)}{N-p}$ and $\alpha>0$. By carefully studying the linearized problem and applying the approximation method and bifurcation theory, we prove that, when the parameter $\al$ takes the critical values $\al(k):=\frac{p\sqrt{(N+p-2)^2+4(k-1)(p-1)(k+N-1)}-p(N+p-2)}{2(p-1)}$ for $k\geq2$, the above quasi-linear H\'enon equation admits non-radial solutions $u$ such that $u\sim |x|^{-\frac{N-p}{p-1}}$ and $|\nabla u|\sim |x|^{-\frac{N-1}{p-1}}$ at $\infty$. One should note that, $\alpha(k)=2(k-1)$ for $k\geq2$ when $p=2$. Our results successfully extend the classical work of F. Gladiali, M. Grossi, and S. L. N. Neves in \cite{GGN} concerning the Laplace operator (i.e., the case $p=2$) to the more general setting of the nonlinear $p$-Laplace operator ($1<p<N$). We overcome a series of crucial difficulties, including the nonlinear feature of the $p$-Laplacian $\Delta_p$, the absence of Kelvin type transforms and the lack of the Green integral representation formula.