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I am interested in positive ground state solutions of the following elliptic pde:

$-\Delta u(x) = u(x)^{p-\epsilon} $ in the unit ball $B$ in $ R^N$ with $ u=0$ on $ \partial B$. Here $ p:=\frac{N+2}{N-2}$ and $ \epsilon>0$ is small.

Question: what is the behaviour of $ u_\epsilon(r)$ as $ \epsilon \searrow 0$ for $ r$ near zero.

Thanks for the replies.

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  • $\begingroup$ Do you want some a priori estimate, and then pass to the limit to prove the existence for the weak solution for the critical power $p=\frac{N+2}{N-2}$? For the variational approach of Yamabe problem or prescribing scalar curvature problem, the strategy is to pass from the subcritical case to the critical case, and to claim that blowing up can not happen in some sense. $\endgroup$
    – Shaoming Guo
    Commented Feb 28, 2012 at 23:10
  • $\begingroup$ I am certain that blow-up does occur (since we know there is no positive solution to the limit problem). I tried rescaling and then using the fact that one knows the explicit solution of $ -\Delta v(x)= v(x)^p$ in $ R^N$ but the result this is giving me seems a bit weird... thanks. $\endgroup$
    – david
    Commented Feb 29, 2012 at 3:39

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