On principal eigenpairs for the (p,q)-Laplacian in exterior domain

We consider an eigenvalue problem of the form \begin{equation*} \left{\begin{array}{rclll} -Δ_{p} u -Δ_{q} u&=& λK(x)|u|^{p-2}u & \mbox{ in } Ω^e u&=&0\qquad \quad &\mbox{ on } \partial Ω u(x) &\to& 0 &\mbox{ as } |x| \to \infty,, \end{array}\right. \end{equation*} where ${\mathrm{\Omega }}^e$ is the exterior of a simply connected, bounded domain $\mathrm{\Omega }$ in ${\mathbb{R}}^N$, $p,q\in (1,N)$ with $p\mathrm{\ne }q$, $0<K\in L^{\mathrm{\infty }}({\mathrm{\Omega }}^e)\cap L^{\frac{N}{p}}({\mathrm{\Omega }}^e)$, and $\lambda \in \mathbb{R}$. We establish the existence of an unbounded set of the principal eigenvalues and corresponding eigenfunctions. Moreover, we establish the regularity, positivity and the asymptotic profiles of these eigenfunctions with respect to the eigenvalue parameter $\lambda$. We use the {\em fibering method} of S.~I. Pohozaev to prove our results.