We consider the following weighted eigenvalue problem in the exterior domain: -Δpu=λK(x)|u|p-2uinB1c,u=0on∂B1,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _pu = \lambda &{}K(x) |u|^{p-2}u \quad \mathrm{in} \quad {B_1^c},\\ u = 0 &{}\quad \mathrm{on}\quad \partial B_1, \end{array}\right. } \end{aligned}$$\end{document}where Δp\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _p$$\end{document} is the p\documentclass[12pt]{minimal}
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\begin{document}$$p$$\end{document}-Laplace operator with p>1,\documentclass[12pt]{minimal}
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\begin{document}$$p>1,$$\end{document} and B1c\documentclass[12pt]{minimal}
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\begin{document}$${B_1^c}$$\end{document} is the exterior of the closed unit ball in RN\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^N$$\end{document} with N≥1.\documentclass[12pt]{minimal}
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\begin{document}$$N\ge 1.$$\end{document} There is no restriction on the dimension N\documentclass[12pt]{minimal}
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\begin{document}$$N$$\end{document} in terms of p,\documentclass[12pt]{minimal}
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\begin{document}$$p,$$\end{document} i.e., we allow both 1<p<N\documentclass[12pt]{minimal}
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\begin{document}$$1< p< N$$\end{document} and p≥N\documentclass[12pt]{minimal}
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\begin{document}$$p\ge N$$\end{document}. The weight function K\documentclass[12pt]{minimal}
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\begin{document}$$K$$\end{document} is locally integrable on B1c\documentclass[12pt]{minimal}
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\begin{document}$${B_1^c}$$\end{document} and is allowed to change its sign. For some appropriate choice of w\documentclass[12pt]{minimal}
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\begin{document}$$w$$\end{document}, a positive weight function on the interval (1,∞)\documentclass[12pt]{minimal}
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\begin{document}$$(1,\infty )$$\end{document}, we prove that the Beppo-Levi space D01,p(B1c)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal D}^{1,p}_0(B_1^c)}$$\end{document} is compactly embedded into the weighted Lebesgue space Lp(B1c;w(|x|)).\documentclass[12pt]{minimal}
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\begin{document}$$L^p({B_1^c};w(|x|)).$$\end{document} The existence of the positive eigenvalue for the above problem is proved for K\documentclass[12pt]{minimal}
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\begin{document}$$K$$\end{document} such that suppK+\documentclass[12pt]{minimal}
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\begin{document}$$K^+$$\end{document} is of non-zero measure and |K|≤w\documentclass[12pt]{minimal}
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\begin{document}$$ |K| \le w$$\end{document}. Further, we discuss the positivity, the regularity and the asymptotic behaviour at infinity of the first eigenfunctions.
