Counterexample Sobolev inequality?

Question

Below is the Gagliardo-Nirenberg-Sobolev inequality I am using:

Let $u\in C_c^\infty(\mathbb R^n\to\mathbb R$) for $n>1$ and $1\leq p <n$ then there exists $C>0$ independent of $u$ such that $$\|u\|_{L^q(\mathbb R^n)}\leq C\|Du\|_{L^p(\mathbb R^n)},$$ where $q=\frac{np}{n-p}$.

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Could somebody please explain the flaw in my thinking that this is a counterexample!

Define the sequence of functions $f_m\in C_c^\infty(\mathbb R^n)$ by $$f_m(x)=1, |x|\leq m\\ f_m(x)=0, |x|\geq m+1$$ and $f_m$ smooth for $|x|\in(m,m+1)$ with the uniform bound $\|Df_m\|_\infty\leq k$. Then $\|f_m\|_q\to\infty$ as $m\to\infty$ yet $\|Df_m\|_p$ remains bounded.

Answer 1

The issue is that $||Df_m||_p$ does not remain bounded.

$Df_m$ is non-zero on the domain $\{x|m\leq |x|\leq m+1\}$, the size of which tends to infinity as $m$ tends to infinity.