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Mixed eigenvalues of discrete \(p\)-Laplacian. (English) Zbl 1336.39004

Summary: This paper deals with the principal eigenvalue of discrete \(p\)-Laplacian on the set of nonnegative integers. Alternatively, it is studying the optimal constant of a class of weighted Hardy inequalities. The main goal is the quantitative estimates of the eigenvalue. The paper begins with the case having reflecting boundary at origin and absorbing boundary at infinity. Several variational formulas are presented in different formulation: the difference form, the single summation form, and the double summation form. As their applications, some explicit lower and upper estimates, a criterion for positivity (which was known years ago), as well as an approximating procedure for the eigenvalue are obtained. Similarly, the dual case having absorbing boundary at origin and reflecting boundary at infinity is also studied. Two examples are presented at the end of Section 2 to illustrate the value of the investigation.

MSC:

39A12 Discrete version of topics in analysis
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35P15 Estimates of eigenvalues in context of PDEs
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References:

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