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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.

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I suggest you write out the Euler-Lagrange equation (which is the p-Laplace equation), and then write the equation in spherical coordinates. Rotationally symmetric solutions mean that the function is of $r$ alone, which should give you an ODE. – Ray Yang Feb 10 '13 at 19:54

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