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

Abstract

In this paper, we investigate the following $D^{1,p}$-critical quasi-linear Hénon equation involving $p$-Laplacian \begin{equation*}\label{00} \left\{ \begin{aligned} &-Δ_p u=|x|^αu^{p_\al^*-1}, & x\in \R^N, \\ &u>0, & x\in \R^N, \end{aligned} \right. \end{equation*} where $N\ge 2$, $1<p<N$, $p_\al^*:=\frac{p(N+\al)}{N-p}$ 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 $k\ge 2$, the above quasi-linear Hénon 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 $k\ge 2$ 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.