Let $V(x)=x_1^2x_2^2 $ be a polynomial defined on $\mathbb{R}^{2}$, we denote $A_V(x)=\sum\limits_{2\le |\alpha|\le 4}|\partial^{\alpha}_xV(x)|^{\frac{1}{|\alpha|}}$.
Please help me to find an $0<\epsilon<\frac{3}{4}$ such that the following result is true:
There exists a constant $c>0$ such that for all $x,x_0\in \mathbb{R}^2$ $$|x-x_0|\le \frac{1}{\Big(A_V(x)\Big)^{\frac{1}{4}+\epsilon}}\, \text{ implies }\,\left(\frac{\Big(A_V(x)\Big)^{\frac{1}{4}+\epsilon}}{\Big(A_V(x_0)\Big)^{\frac{1}{4}+\epsilon}}\right)^{\pm 1}\le c$$