Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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EXISTENCE OF GLOBAL SOLUTIONS TO SOME NONLINEAR EQUATIONS ON LOCALLY FINITE GRAPHS

  • Received : 2020.04.28
  • Accepted : 2020.11.12
  • Published : 2021.05.01
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Abstract

Let G = (V, E) be a connected locally finite and weighted graph, ∆p be the p-th graph Laplacian. Consider the p-th nonlinear equation -∆pu + h|u|p-2u = f(x, u) on G, where p > 2, h, f satisfy certain assumptions. Grigor'yan-Lin-Yang [24] proved the existence of the solution to the above nonlinear equation in a bounded domain Ω ⊂ V. In this paper, we show that there exists a strictly positive solution on the infinite set V to the above nonlinear equation by modifying some conditions in [24]. To the m-order differential operator 𝓛m,p, we also prove the existence of the nontrivial solution to the analogous nonlinear equation.

Keywords

Acknowledgement

The first author is supported by National Natural Science Foundation of China under Grant No. 11971053. The second author is supported by National Natural Science Foundation of China under Grant No. 11871094.

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