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I am starting from Theorem 3.4 in Struwe's book Variational methods. The authors proves an existence result for a problem with critical growth. I would like to replace the laplacian with the fractional laplacian $(-\Delta)^s$, with $0<s<N/2$. In particular, the equation becomes $(-\Delta)^s u = u^{\frac{N+2}{N-2s}}$, $u>0$ in $\Omega$, with $u=0$ on $\partial \Omega$. Struwe's family $u_t^\sigma$ becomes, for $t \in (0,1)$ and $\sigma \in S^{N-1}$, $$ u_t^\sigma (x) = \left( \frac{1-t}{(1-t)^2+|x-t\sigma |^2} \right)^{\frac{N-2s}{2}}, $$ which is the multiplied by the cut-off $\varphi_R$ and normalized to get $$v_t^\sigma = \frac{\varphi_R u_t^\sigma}{\|\varphi_R u_t^\sigma \|_{L^{\frac{2N}{N-2s}}} }$$ exactly as in Struwe's book.

Now my question is: define the fractional barycenter by $F(u) = \int_\Omega x |(-\Delta)^{\frac{s}{2}}u|^2\, dx$. Is it true that $$\lim_{t \to 1} F(v_t^\sigma) = \sigma?$$ In the local case, this follows easily from Leibnitz's rule of differentiation. In the fractional case I am completely stuck.

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  • $\begingroup$ Just one curiosity: Why do you call $$\int_\Omega x |(-\Delta)u|^2 dx$$ a barycenter? If $u$ is a density, should not the barycenter be $$\int_\Omega x u\; dx$$ instead? $\endgroup$ Oct 28, 2013 at 14:21

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