AI 中文总结
研究拟线性加权N-拉普拉斯刘维尔方程，通过研究线性化问题、应用逼近方法和分歧理论，在特定参数条件下证明存在非径向解，推广了相关存在性结果，克服了诸多关键困难。
AI 中文摘要
本文研究如下拟线性加权N-拉普拉斯刘维尔方程：\(-\Delta_N u = |x|^{N\alpha}e^{u}, x\in \mathbb{R}^N\)，其中\(N \geq 2\)。对于\(\alpha>0\)，通过仔细研究线性化问题并应用逼近方法和分歧理论，证明当参数\(\alpha\)等于临界值\(\alpha(k):=\frac{\sqrt{k(N - 1)(k + N - 2)}}{N - 1}-1\)（\(k \geq 2\)）时，存在非径向解\(u\)（从\(U_{\alpha(k)}\)分歧出来），使得\(u\sim \ln|x|\)，\(|\nabla u| = O(|x|^{-1})\)在无穷远处成立，且\(\int_{\mathbb{R}^N}|x|^{N\alpha}e^{u}\mathrm{d}x=N\left(\frac{N^2}{N - 1}\right)^{N - 1}(\alpha + 1)^{N - 1}\omega_N\)。结果成功将二维拉普拉斯情形的存在性结果推广到更一般的N维N-拉普拉斯情形，克服了包括N-拉普拉斯非线性性质等一系列关键困难。
英文摘要
In this paper, we investigate the following quasi-linear weighted $N$-Laplacian Liouville equation \begin{equation*}\label{0} -Δ_N u=|x|^{Nα}e^{u}, \qquad x\in \R^N, \end{equation*} where $N \geq 2$. For $\al>0$, by carefully studying the linearized problem and applying the approximation method and bifurcation theory, we prove that, when the parameter $α$ equals to the critical values $α(k):=\frac{\sqrt{k(N-1)(k+N-2)}}{N-1}-1$ for $k \geq 2$, there exist non-radial solutions $u$ (bifurcating from $U_{α(k)}$) to the above quasi-linear Hénon type Liouville equation such that $u\sim \ln|x|$, $|\nabla u|= O(|x|^{-1})$ at $\infty$ and $\int_{\R^N}|x|^{Nα}e^{u}\md x=N\left(\frac{N^2}{N-1}\right)^{N-1}(α+1)^{N-1}ω_N$. One should note that, $α(k)=k-1$ for $k\geq2$ when $N=2$. Our results successfully extend the existence result of J. Prajapat and G. Tarantello in \cite{PT} concerning the $2$-dimension and Laplacian case (i.e., $N=2$) to the more general $N$-dimension and $N$-Laplacian cases ($N\geq 2$), and extend the results of F. Gladiali, M. Grossi, and S. L. N. Neves in \cite{GGN} and the authors in \cite{DDGL} from $1<p<N$ to the much more complicated limiting case $p=N$. We introduced some new ideas and overcame a series of crucial difficulties, including the nonlinearity nature of the $N$-Laplacian $Δ_N$, the lack of Green integral representation formula and critical weighted Sobolev embedding inequality, the absence of Kelvin type transforms for linearized/difference equations, the invariance of the total mass under scalings of $u$, and the signs-changing and divergence (to $-\infty$) at $\infty$ of the solutions, which makes the suitable choices of the approximate problems, the (normalized) approximate function sequences and the working space to be quite difficult.
