Existence of continuous eigenvalues for a class of parametric problems involving the (p,2)-laplacian operator
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 bound...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Springer
2019
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| Online Access: | https://eprints.nottingham.ac.uk/56225/ |
| _version_ | 1848799297101889536 |
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| author | Bhattacharya, Tilak Emamizadeh, Behrouz Farjudian, Amin |
| author_facet | Bhattacharya, Tilak Emamizadeh, Behrouz Farjudian, Amin |
| author_sort | Bhattacharya, Tilak |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T20:33:25Z |
| format | Article |
| id | nottingham-56225 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T20:33:25Z |
| publishDate | 2019 |
| publisher | Springer |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-562252020-02-07T04:30:12Z https://eprints.nottingham.ac.uk/56225/ Existence of continuous eigenvalues for a class of parametric problems involving the (p,2)-laplacian operator Bhattacharya, Tilak Emamizadeh, Behrouz Farjudian, Amin 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. Springer 2019-02-07 Article PeerReviewed application/pdf en https://eprints.nottingham.ac.uk/56225/1/2019-Bhattacharya_Emamizadeh_Farjudian-Continuous_Eigenvalues.pdf 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 Fibering method; Continuous eigenvalues; p-Laplacian http://dx.doi.org/10.1007/s10440-019-00241-9 doi:10.1007/s10440-019-00241-9 doi:10.1007/s10440-019-00241-9 |
| spellingShingle | Fibering method; Continuous eigenvalues; p-Laplacian Bhattacharya, Tilak Emamizadeh, Behrouz Farjudian, Amin Existence of continuous eigenvalues for a class of parametric problems involving the (p,2)-laplacian operator |
| title | Existence of continuous eigenvalues for a class of parametric problems involving the (p,2)-laplacian operator |
| title_full | Existence of continuous eigenvalues for a class of parametric problems involving the (p,2)-laplacian operator |
| title_fullStr | Existence of continuous eigenvalues for a class of parametric problems involving the (p,2)-laplacian operator |
| title_full_unstemmed | Existence of continuous eigenvalues for a class of parametric problems involving the (p,2)-laplacian operator |
| title_short | Existence of continuous eigenvalues for a class of parametric problems involving the (p,2)-laplacian operator |
| title_sort | existence of continuous eigenvalues for a class of parametric problems involving the (p,2)-laplacian operator |
| topic | Fibering method; Continuous eigenvalues; p-Laplacian |
| url | https://eprints.nottingham.ac.uk/56225/ https://eprints.nottingham.ac.uk/56225/ https://eprints.nottingham.ac.uk/56225/ |