2026/07/27 by Wei Dai, Lixiu Duan, Changfeng Gui +1
#math.AP
In this paper, we investigate the following quasi-linear weighted N-Laplacian Liouville equation -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|Nαeu\md 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 \citePT 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 \citeGGN and the authors in \citeDDGL 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.