# Mathematische Nachrichten

## Low dimensional instability for quasilinear problems of weighted exponential nonlinearity

### Abstract

We prove a sharp Liouville type theorem for stable ${W}_{loc}^{1,p}$ solutions of equation

on the entire Euclidean space ${\mathbb{R}}^{N}$, where $p>2$ and f is a continuous and nonnegative function in ${\mathbb{R}}^{N}\setminus \left\{0\right\}$ such that $f\left(x\right)\ge {a|x|}^{q}$ as $|x|\ge {R}_{0}>0$, where $q>-p$ and $a>0$. Our theorem holds true for $2\le N<\frac{{p}^{2}\phantom{\rule{0.16em}{0ex}}+\phantom{\rule{0.16em}{0ex}}3p\phantom{\rule{0.16em}{0ex}}+\phantom{\rule{0.16em}{0ex}}4q}{p\phantom{\rule{0.16em}{0ex}}-\phantom{\rule{0.16em}{0ex}}1}$ and is sharp in the case $f\left(x\right)={a|x|}^{q}$.

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