Commun. Korean Math. Soc. 2011; 26(4): 591-601
Printed December 1, 2011
https://doi.org/10.4134/CKMS.2011.26.4.591
Copyright © The Korean Mathematical Society.
Soon-Yeong Chung and Heesoo Lee
Sogang University, Sogang University
In this paper, we deal with the discrete $p$-Laplacian operators with a potential term having the smallest nonnegative eigenvalue. Such operators are classified as its smallest eigenvalue is positive or zero. We discuss differences between them such as an existence of solutions of $p$-Laplacian equations on networks and properties of the energy functional. Also, we give some examples of Poisson equations which suggest a difference between linear types and nonlinear types. Finally, we study characteristics of the set of a potential those involving operator has the smallest positive eigenvalue.
Keywords: discrete Laplacian, nonlinear elliptic equations
MSC numbers: 35J60
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