Let $u\in \mathcal C^1_c(\mathbb R^n)$ and let $D$ it's support. show that there is a function $v\in \mathcal C^1_c(\mathbb R^n)$ of the form $v(x)=\lambda ^\alpha u(k^\beta x)$ s.t. $$\|v\|_{L^q}=\|v\|_{W^{1,p}}=1\ \ ?$$
Attempt
Let $$\int_D |v|^q=C,$$ I can set $K=\frac{1}{C^{\frac{1}{nq}}}D$ and thus get $$\int_D |u|^q=C\int_K u\left(C^\frac{1}{nq} y\right)dy,$$ and thus $v(y)=u(C^{\frac{1}{nq}}y)$ has norm $L^q$ of $1$. But how can it be adapted such that $\|v\|_{W^{1,p}(K)}=1$ ?
Sketch- Lets simplify notation, so $v= a u(bx)$ for constants $a,b>0$ to be determined. Compute the norms of $v$ in terms of $u$. You should see that $$ ‖v‖_{L^p} = a b^{-n} ‖u‖_{L^p} $$ Also by chain rule $$ ‖\partial_i v‖_{L^p} = ab^{-n+1}‖\partial_i u‖_{L^p}$$ Hence(this step depends on your definition of the norm, if you have a different one then the exact expression will be different), $$‖v‖_{W^{1,p}} = ab^{-n} ‖u‖_{L^p} + ab^{-n+1}‖∇ u‖_{L^p}$$
From here it should be easy to conclude that it can be done, and $a,b$ can be computed.