Existence of Continuous Eigenvalues for a Class of Parametric Problems Involving the \((p,2)\)-Laplacian Operator
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We discuss a parametric eigenvalue problem, where the differential operator is of \((p,2)\)-Laplacian type. We show that, when \(p\neq 2\), 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 \(p < 2\).
KeywordsFibering method Continuous eigenvalues \(p\)-Laplacian
Mathematics Subject Classification (2010)35J60 35P30
The authors wish to thank the anonymous referee for their corrections and constructive suggestions.
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