Existence of continuous eigenvalues for a class of parametric problems involving the (p,2)-laplacian operator

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Bhattacharya, Tilak, Emamizadeh, Behrouz and Farjudian, Amin (2019) Existence of continuous eigenvalues for a class of parametric problems involving the (p,2)-laplacian operator. Acta Applicandae Mathematicae . pp. 1-15. ISSN 0167-8019

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Official URL: http://dx.doi.org/10.1007/s10440-019-00241-9

Abstract

We discuss a parametric eigenvalue problem, where the differential operator is of (p,2)-Laplacian type. We show that, when p≠2, the spectrum of the operator is a half line, with the end point formulated in terms of the parameter and the principal eigenvalue of the Laplacian with zero Dirichlet boundary conditions. Two cases are considered corresponding to p>2 and p<2, and the methods that are applied are variational. In the former case, the direct method is applied, whereas in the latter case, the fibering method of Pohozaev is used. We will also discuss a priori bounds and regularity of the eigenfunctions. In particular, we will show that, when the eigenvalue tends towards the end point of the half line, the supremum norm of the corresponding eigenfunction tends to zero in the case of p>2, and to infinity in the case of p<2.

Item Type:

Article

Additional Information:

“This is a post-peer-review, pre-copyedit version of an article published in Acta Applicandae Mathematicae. The final authenticated version is available online at: https://doi.org/10.1007/s10440-019-00241-9”. © Springer Nature B.V. 2019

Keywords:

Fibering method; Continuous eigenvalues; p-Laplacian

Schools/Departments:

University of Nottingham Ningbo China > Faculty of Science and Engineering > School of Computer Science
University of Nottingham Ningbo China > Faculty of Science and Engineering > School of Mathematical Sciences

Identification Number:

https://doi.org/10.1007/s10440-019-00241-9

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