Non-radial solutions for the critical quasi-linear H\'{e}non equation involving $p$-Laplacian in $\R^N$

TL;DR

This paper finds non-radial solutions to a critical quasi-linear Hénon equation involving the p-Laplacian in ^N, extending classical results from the Laplacian case to the nonlinear p-Laplacian setting.

Contribution

It introduces new methods to establish non-radial solutions for the p-Laplacian Hfnen equation at critical parameters, overcoming key nonlinear and analytical challenges.

Findings

Existence of non-radial solutions at critical ff values of ff.

Extension of classical Laplacian results to p-Laplacian case.

Identification of asymptotic behaviors of solutions at infinity.

Abstract

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.