Weighted quasilinear eigenvalue problems in exterior domains

Abstract

We consider the following weighted eigenvalue problem in the exterior domain:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _pu = \lambda &{}K(x) |u|^{p-2}u \quad \mathrm{in} \quad {B_1^c},\\ u = 0 &{}\quad \mathrm{on}\quad \partial B_1, \end{array}\right. } \end{aligned}$$

where $\Delta _p$ is the $p$-Laplace operator with $p>1,$ and ${B_1^c}$ is the exterior of the closed unit ball in $\mathbb {R}^N$ with $N\ge 1.$ There is no restriction on the dimension $N$ in terms of $p,$ i.e., we allow both $1< p< N$ and $p\ge N$. The weight function $K$ is locally integrable on ${B_1^c}$ and is allowed to change its sign. For some appropriate choice of $w$, a positive weight function on the interval $(1,\infty )$, we prove that the Beppo-Levi space ${{\mathcal D}^{1,p}_0(B_1^c)}$ is compactly embedded into the weighted Lebesgue space $L^p({B_1^c};w(|x|)).$ The existence of the positive eigenvalue for the above problem is proved for $K$ such that supp$K^+$ is of non-zero measure and $|K| \le w$. Further, we discuss the positivity, the regularity and the asymptotic behaviour at infinity of the first eigenfunctions.