A weighted eigenvalue problem for mixed local and nonlocal operators with potential

Radhakrishnan Lakshmi, Ratan Kr. Giri and Sekhar Ghosh

Mathematische Nachrichten, 2026, vol. 299, issue 2, 367-396

Abstract: We study an indefinite weighted eigenvalue problem for an operator of mixed‐type (that includes both the classical p$p$‐Laplacian and the fractional p$p$‐Laplacian) in a bounded open subset Ω⊂RN(N≥2)$\Omega \subset \mathbb {R}^N \,(N\ge 2)$ with Lipschitz boundary ∂Ω$\partial \Omega$, which is given by −Δpu+(−Δp)su+V(x)|u|p−2u=λg(x)|u|p−2uinΩ,u=0inRN∖Ω,$$\begin{align*} -\Delta _p u + (-\Delta _p)^su+V(x)|u|^{p-2}u&=\lambda g(x)|u|^{p-2}u\nobreakspace \text{in}\nobreakspace \Omega,\cr u&=0\nobreakspace \text{in}\nobreakspace \mathbb {R}^N\setminus \Omega, \end{align*}$$where λ>0$\lambda >0$ is a parameter, exponents 0 0$V\ge 0, g > 0$ a.e. in Ω$\Omega$. Using the variational tools together with a weak comparison and strong maximum principles, we investigate the existence and uniqueness of principal eigenvalue and discuss its qualitative properties. Moreover, with the help of Ljusternik–Schnirelman category theory, it is proved that there exists a nondecreasing sequence of positive eigenvalues, which goes to infinity. Further, we show that the set of all positive eigenvalues is closed, and eigenfunctions associated with every positive eigenvalue are bounded.

Date: 2026
References: Add references at CitEc
Citations:

Downloads: (external link)
https://doi.org/10.1002/mana.70093

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:bla:mathna:v:299:y:2026:i:2:p:367-396

Ordering information: This journal article can be ordered from
http://www.blackwell ... bs.asp?ref=0025-584X

Access Statistics for this article

Mathematische Nachrichten is currently edited by Robert Denk

More articles in Mathematische Nachrichten from Wiley Blackwell
Bibliographic data for series maintained by Wiley Content Delivery ().