Article contents
On generalized eigenvalue problems of fractional (p, q)-Laplace operator with two parameters
Published online by Cambridge University Press: 22 January 2024
Abstract
For $s_1,\,s_2\in (0,\,1)$ and $p,\,q \in (1,\, \infty )$
, we study the following nonlinear Dirichlet eigenvalue problem with parameters $\alpha,\, \beta \in \mathbb {R}$
driven by the sum of two nonlocal operators:
is a bounded open set. Depending on the values of $\alpha,\,\beta$
, we completely describe the existence and non-existence of positive solutions to (P). We construct a continuous threshold curve in the two-dimensional $(\alpha,\, \beta )$
-plane, which separates the regions of the existence and non-existence of positive solutions. In addition, we prove that the first Dirichlet eigenfunctions of the fractional $p$
-Laplace and fractional $q$
-Laplace operators are linearly independent, which plays an essential role in the formation of the curve. Furthermore, we establish that every nonnegative solution of (P) is globally bounded.
Keywords
- Type
- Research Article
- Information
- Copyright
- Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh
Footnotes
This article has been updated since it was orignially published. A notice detailing this has been published and the errors rectified in the online PDF and HTML copies.
References
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