Non-radial solutions for the quasi-linear Hénon type N-Laplacian Liouville equation
Abstract
In this paper, we investigate the following quasi-linear weighted N-Laplacian Liouville equation label0 −ΔN u=|x|Nαeu, x∈ RN, where N ≥ 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):=(√(k(N−1)(k+N−2)))/(N−1)−1 for k ≥ 2, there exist non-radial solutions u (bifurcating from Uα(k)) to the above quasi-linear Hénon type Liouville equation such that u∼ ln|x|, |∇ u|= O(|x|−1) at ∞ and ∫RN|x|Nαeumd x=N(N2/(N−1))N−1(α+1)N−1ωN. One should note that, α(k)=k−1 for k≥2 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≥ 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 ΔN, 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 −∞) at ∞ 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.
Related
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BibTeX
@misc{dai2026nonradial,
title = {{Non-radial solutions for the quasi-linear Hénon type $N$-Laplacian Liouville equation}},
author = {Wei Dai and Lixiu Duan and Changfeng Gui and Yuan Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.24012},
eprint = {2607.24012},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.24012},
}
RIS
TY - UNPB
TI - Non-radial solutions for the quasi-linear Hénon type N-Laplacian Liouville equation
AU - Wei Dai
AU - Lixiu Duan
AU - Changfeng Gui
AU - Yuan Li
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2607.24012
UR - https://arxiv.org/abs/2607.24012
AB - In this paper, we investigate the following quasi-linear weighted N-Laplacian Liouville equation label0 −Δ_N u=|x|^Nαe^u, x∈ R^N, where N ≥ 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):=(√(k(N−1)(k+N−2)))/(N−1)−1 for k ≥ 2, there exist non-radial solutions u (bifurcating from U_α(k)) to the above quasi-linear Hénon type Liouville equation such that u∼ ln|x|, |∇ u|= O(|x|^−1) at ∞ and ∫_R^N|x|^Nαe^umd x=N(N^2/(N−1))^N−1(α+1)^N−1ω_N. One should note that, α(k)=k−1 for k≥2 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≥ 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 −∞) at ∞ 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.
ER -