Let G = (V,E) be a finite connected weighted graph, and assume 1 ⩽ α ⩽ p ⩽ q. In this paper, we consider the p-th Yamabe type equation ―∆pu+huq―1 = λfuα―1 on G, where ∆p is the p-th discrete graph Laplacian, h < 0 and f > 0 are real functions defined on all vertices of G. Instead of H. Ge’s approach [Proc. Amer. Math. Soc., 2018, 146(5): 2219–2224], we adopt a new approach, and prove that the above equation always has a positive solution u > 0 for some constant λ ∈ ℝ. In particular, when q = p, our result generalizes Ge’s main theorem from the case of α ⩾ p > 1 to the case of 1 ⩽ α ⩽ p, It is interesting that our new approach can also work in the case of α ⩾ p > 1.
p-th Yamabe type equation graph Laplacian positive solutions
34B35 35B15 58E30
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The first author would like to thank Professor Yanxun Chang for constant guidance and encouragement. The second author would like to thank Professor Gang Tian and Huijun Fan for constant encouragement and support. Both authors would also like to thank Professor Huabin Ge for many helpful conversations. The first author was supported by the National Natural Science Foundation of China (Grant Nos. 11471138, 11501027, 11871094) and the Fundamental Research Funds for the Central Universities (Grant No. 2017JBM072). The second author was supported by the National Natural Science Foundation of China (Grant No. 11401578).
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