| Contents |
| We study the existence of positive solutions to problems of the form
\begin{equation*}
\label{eqn4}
\left\{
\begin{split}
-\Delta_p u &= \lambda K(x) f(u) \hspace{.1in}\mbox{in } B_1^c,
\\u &= 0 \hspace{.762in} \mbox{on } \partial B_1 ,
\\u(x)&\rightarrow0 \hspace{.75in} \mbox{as }\left|x \right|\rightarrow\infty,
\end{split} \right.
\end{equation*}
where $B_1^c=\{x\in\ \mathbb{R}^{n}\ |~ \left|x \right|> 1\}$, $\Delta_p u=\textrm{div}(|\nabla{u}|^{p-2}\nabla{u}), p \in (1, n)$, $\lambda$ is a positive parameter, $K$ belongs to a class of functions which satisfy certain decay assumptions and $f$ belongs to a class of $(p-1)-$subhomogeneous functions which may be singular at the origin, namely $\lim_{s\rightarrow 0^{+}}f(s) =-\infty$. We use the method of sub and super solutions to prove our result. Our methods can be also applied to establish a similar existence result when the domain is entire $\mathbb{R}^{n}$. |
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