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On generalized eigenvalue problems of fractional (p, q)-Laplace operator with two parameters

Proceedings of the Royal Society of Edinburgh Section A: Mathematics, ISSN: 1473-7124, Page: 1-46
2024
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Article Description

For and, we study the following nonlinear Dirichlet eigenvalue problem with parameters driven by the sum of two nonlocal operators: where is a bounded open set. Depending on the values of, we completely describe the existence and non-existence of positive solutions to (P). We construct a continuous threshold curve in the two-dimensional -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 -Laplace and fractional -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.

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