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On the Elliptic Equation $\triangle u + |x|^\ell K(x)u^{p^*}=0$ in $R^n$

Jann-Long Chern

In this paper, we consider the elliptic equation $\triangle u +
|x|^\ell K(x) u^{p^*}=0$  in  $R^n, n\ge 3$, where $\ell > -2$, $K$ is
a suitable function on $R^n$ and $p^*=\frac {n+2+2\ell}{n-2}$.  We
obtain a necessary and sufficient condition for the existence of a
positive radial solution $u=u(|x|)$ with the asymptotic behavior
$\lim\limits_{r\to \infty} r^{n-2} u(r)=\b > 0$.  This condition makes
contact with the Kazdan-Warner obstruction.  We also give several
sufficient conditions for solvability extending several earlier
results.

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