Positive solutions of p-th Yamabe type equations on graphs

Abstract

Let G = (V,E) be a nite connected weighted graph, and assume \mathrm{1}\le \alpha \le p\le q. In this paper, we consider the p-th Yamabe type equation -{\Delta p}_{}u+{huq-\mathrm{1}}^{}={\lambda fu\alpha -\mathrm{1}}^{} on G, where {\Delta p}_{} is the p-th discrete graph Laplacian, h<0 and f>0 are real functions dened 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 \lambda \in ℝ. In particular, when q = p; our result generalizes Ge's main theorem from the case of \alpha \ge p\rangle \mathrm{1} to the case of \mathrm{1}\le \alpha \le p. It is interesting that our new approach can also work in the case of \alpha \ge p\rangle \mathrm{1}.

Key words： p-th Yamabe type equation; graph Laplacian; positive solutions

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