I have the the following problem and I just can't get my head around how to solve it. Be $1<p<n$ and $q=\frac{np}{n-p}$, $u\in\mathcal{C}_{n,p}=\{f\in W^{1,p}_{loc}: \|f\|_{L^q(\mathbb{R}^n)}=1\}$ and
$ \mathcal{F}(u)=\int_{\mathbb{R}^n}|Du|^p. $
Find all positive, rotational symmetric solutions for the corresponding Euler-Lagrange-Equation.
I don't need a perfect solution just some ideas on how to solve this.