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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.27701 |
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| _version_ | 1866910195078135808 |
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| author | Chhetri, Maya Drabek, Pavel Shivaji, Ratnasingham |
| author_facet | Chhetri, Maya Drabek, Pavel Shivaji, Ratnasingham |
| contents | 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 $Ω^e$ is the exterior of a simply connected, bounded domain $Ω$ in $\mathbb{R}^N$, $p, q \in (1, N)$ with $p \neq q$, $0 < K \in L^{\infty}(Ω^e) \cap L^{\frac{N}{p}}(Ω^e)$, and $λ\in \mathbb{R}$. 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 $λ$. We use the {\em fibering method} of S.~I. Pohozaev to prove our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27701 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On principal eigenpairs for the (p,q)-Laplacian in exterior domain Chhetri, Maya Drabek, Pavel Shivaji, Ratnasingham Analysis of PDEs 35J25, 35J60, 35J62 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 $Ω^e$ is the exterior of a simply connected, bounded domain $Ω$ in $\mathbb{R}^N$, $p, q \in (1, N)$ with $p \neq q$, $0 < K \in L^{\infty}(Ω^e) \cap L^{\frac{N}{p}}(Ω^e)$, and $λ\in \mathbb{R}$. 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 $λ$. We use the {\em fibering method} of S.~I. Pohozaev to prove our results. |
| title | On principal eigenpairs for the (p,q)-Laplacian in exterior domain |
| topic | Analysis of PDEs 35J25, 35J60, 35J62 |
| url | https://arxiv.org/abs/2603.27701 |