Let $$ F(x)=\frac{1}{2}\|x\|_{l^p(\mathbb{R}^n)}^2=\frac{1}{2}\Big(\sum_{i=1}^{n}|x_i|^p\Big)^\frac{2}{p}, $$ for $x=(x_1,x_2,\ldots,x_n)\in\mathbb{R}^n$. Is the gradient of $F(x)$, i.e. $G(x)=\nabla F(x)$ Lipschitz continuous in $\mathbb{R}^n$?
Indeed the function $G$ will look like below: $$ G(x)=(\sum_{j=1}^{m}|x_j|^{p}\Big)^\frac{2-p}{p}(|x_1|^{p-2}x_1,\ldots,|x_n|^{p-2}x_n). $$ I can only see that for $p=2$, since $G(x)=x$ for $p=2$ and hence Lipschitz continuous. What about $p>2$? Can someone please help. Thanks.