TY - JOUR

T1 - Existence of Continuous Eigenvalues for a Class of Parametric Problems Involving the (p, 2) -Laplacian Operator

AU - Bhattacharya, Tilak

AU - Emamizadeh, Behrouz

AU - Farjudian, Amin

N1 - Publisher Copyright:
© 2019, Springer Nature B.V.

PY - 2020/2/1

Y1 - 2020/2/1

N2 - 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.

AB - 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.

KW - Continuous eigenvalues

KW - Fibering method

KW - p-Laplacian

UR - http://www.scopus.com/inward/record.url?scp=85061182692&partnerID=8YFLogxK

U2 - 10.1007/s10440-019-00241-9

DO - 10.1007/s10440-019-00241-9

M3 - Article

AN - SCOPUS:85061182692

VL - 165

SP - 65

EP - 79

JO - Acta Applicandae Mathematicae

JF - Acta Applicandae Mathematicae

SN - 0167-8019

IS - 1

ER -