Critical Sobolev exponent

Question

Let $n\in\mathbb{N}$, $1\le p<n$ and $c>0$. Does

$$\lVert f\rVert_{L^q(\mathbb{R}^n)}\le c \lVert\nabla f\rVert_{L^p(\mathbb{R}^n)}\text{ for all }f\in C_0^\infty(\mathbb{R}^n)$$

imply

$$q=\frac{np}{n-p}\text{?}$$

Answer 1

Hint: Use the rescaled function $$f_\lambda(x) = f(\lambda x)$$

in your inequality then let $\lambda\to 0$ and $\lambda\to \infty$. conclude that the only possible of $q$ is $q=\frac{np}{n-p}$