Positive solutions of p-th Yamabe type equations on graphs

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+hu^q―1 = λfu^α―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 ∈ ?. 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.