Liouville type theorem for a system of elliptic inequalities on weighted graphs without (p0)-condition
-
Nguyen Cong Minh
and
Ngoc Huong Nguyen
Abstract
In this paper, we study the existence and nonexistence of solutions of a system of inequalities
where (V, E) is an infinite, connected, locally finite weighted graph, p > 1, q > 1, h1, h2 are positive potential functions and Δ is the standard graph Laplacian. We prove that, under some growth assumptions on weighted volume of balls and the existence of a suitable distance on the graph, any nonnegative solution of the above system must be trivial. We also give an application to the N-dimensional integer lattice graph ℤN and show the sharpness of the obtained result. In particular, our result is a natural extension of the recent result [Monticelli, D. D.—Punzo, F.—Somaglia, J.: Nonexistence results for semilinear elliptic equations on weighted graphs, arXiv:2306.03609, (2023)] from a single inequality to a system of inequalities.
Acknowledgement
The authors would like to thank the anonymous referee for the careful reading and helpful suggestions.
Communicated by Giuseppe Di Fazio
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