(untitled) (Journal of Partial Differential Equations, 2017)
" Let G=(V,E) be a locally finite graph, 03a9 2282 V be a finite connected set, 0394 be the graph Laplacian, and suppose that h : V 2192 R is a function satisfying the coercive condition on 03a9, namely there exists some constant 03b4 203a 0 such that 222b_03a9u(-0394+h)ud03bc 2265 03b4 222b_03a9|2207u|00b2d03bc, quad 2200u:V 2192 R. By the mountain-pass theoremof Ambrosette-Rabinowitz, we prove that for any p 203a 2, there exists a positive solution to -0394u+hu=|u|^p-2u quad ; ; in ; ; 03a9. Using the same method, we prove similar results for the p-Laplacian equations. This partly improves recent results of Grigor'yan-Lin-Yang."