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\begin{abstract}
We consider subcritical situations `a la` Trudinger-Moser for problems of the form
$$
-\Delta u=a(x) f(u)\ \mbox{ in }\Omega,\quad u=0\ \mbox{ on }\partial\Omega=\partial B(0,1),
$$
in the exterior of the unit ball in $\mathbb{R^}2$. Here the nonnegative coefficient $a(x)$ satisfies a natural integrability condition and we show existence of extremal {\it constant-sign solutions} as well as {\it sign-changing solutions} for such problems. This is joint work with H. Tehrani at the University of Nevada Las Vegas.
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