The $p$-Laplacian operators with potential terms
Commun. Korean Math. Soc. 2011 Vol. 26, No. 4, 591-601
https://doi.org/10.4134/CKMS.2011.26.4.591
Published online December 1, 2011
Soon-Yeong Chung and Heesoo Lee
Sogang University, Sogang University
Abstract : 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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