\[
%\large
We study nonnegative radial solutions to the problem
\begin{equation*}
\left\{
\begin{split}
-\Delta u = \lambda K(\left|x \right|) f(u), \quad x \in \Omega
\\u = 0 \quad \qquad \quad \qquad \mbox{if } \left|x \right| = r_0
\\u \rightarrow0 \quad \qquad \quad \qquad \mbox{as } \left|x \right|\rightarrow\infty,
\end{split} \right.
\end{equation*}
where $\lambda$ is a positive parameter, $\Delta u=\mbox{div} \big(\nabla u\big)$ is the Laplacian of $u$,
$\Omega=\{x\in\ \mathbb{R}^{n}; n \textgreater 2, \left|x \right| \textgreater r_0\}$ and $K$ belongs to a class of functions such that $\lim_{r\rightarrow \infty}K(r)=0$. For classes of nonlinearities $f$ that are negative at the origin and sublinear at $\infty$ we discuss existence and uniqueness results when $\lambda$ is large.
\]