Riv. Mat. Univ. Parma, Vol. 14, No. 1, 2023

Prashanta Garain [a] and Alexander Ukhlov [b]

Mixed anisotropic and nonlocal Dirichlet \((p,q)\)-eigenvalue problem

Pages: 217-232
Received: 18 May 2023
Accepted in revised form: 27 July 2023
Mathematics Subject Classification: 35M12, 35J92, 35R11, 35P30, 35A01.
Keywords: Mixed local and nonlocal \(p\)-Laplacian, Anisotropic \(p\)-Laplace operator, Dirichlet eigenvalue problem, existence, regularity.
Authors address:
[a]: Indian Institute of Science Education and Research Berhampur, Department of Mathematical Sciences, Berhampur, Odisha, India
[b]: Ben-Gurion University of the Negev, Department of Mathematics, Beer Sheva, Israel

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Abstract: In this article, we consider an anisotropic and a combination of anisotropic and nonlocal Dirichlet \((p,q)\)-eigenvalue problems.
We establish existence and regularity of eigenfunctions in a bounded domain \(\Omega\subset\mathbb{R}^N\) under the assumption that \(1 < p < \infty\) and \(1 < q < p^{*}\) where \(p^{*}=\frac{N p}{N-p}\) if \(1 < p < N\) and \(p^{*}=\infty\) if \(p\geq N\).

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