| Contents |
| We consider the weighted eigenvalue problem
in the exterior domain
\begin{equation*}
\begin{align}
-\Delta_p u &= \lambda K(x) |u|^{p-2}u \quad \mbox{in } B_1^c ,
\\u &= 0 \quad \mbox{on } \partial B_1,
\end{align} \end{equation*}
where $\Delta_p$ is the $p$-Laplace operator for $p\in(1,\infty),$ and $B_1$ is
the closed unit ball in $\mathbb R^N,$ $N\ge 1.$ We consider both the
cases
$N\in(p,\infty)$ and $N\in[1,p].$
For some appropriate choice of $w \in L^1(1,\infty)$ with $w\geq 0,$
we prove that the Beppo-Levi
space $\mathcal{D}^{1,p}_0(B_1^c)(:=$ the completion of
$\mathcal{C}_c^\infty(B_1^c)$
with
respect to $||\nabla u||_p$ norm) is continuously and compactly embedded into
$L^p(B_1^c;w(|x|)).$ Using this embedding, we prove
the
existence of a positive principal eigenvalue for $K$ such that $K^+\not\equiv
0$ and $|K(x)| \le w(|x|)$. |
|