Liouville Type Theorem for Stable Solutions to Weighted Quasilinear Problems in $\mathbb{R}^{N}$

Abstract

In this paper, we prove the Liouville type theorem for stable $W_{\mathrm{loc}}^{1, p}$ solutions of the weighted quasilinear problem

$$ -\operatorname{div}\left (w_{1}(x)\left (s^{2}+|\nabla u|^{2} \right )^{\frac{{p - 2}}{2}}\nabla u\right )=w_{2}(x)f(u)\quad \text{in}~ \mathbb{R}^{N}, $$

where $s \geq 0$ is a real number, $f(u)$ is either $e^{u}$ or $-e^{\frac{1}{u}}$ and $w_{1} (x),w_{2} (x) \in L_{\text{loc}}^{1} \left (\mathbb{R}^{N} \right )$ be nonnegative functions so that $w_{1} (x) \leq C_{1}|x|^{m}$ and $w_{2} (x) \geq C_{2}|x|^{n}$ when $|x|$ is big enough. Here we need $n>m$.