Semi-linear Elliptic Equations on Graph

Abstract

" Let G=(V,E) be a locally finite graph, \u03a9 \u2282 V be a finite connected set, \u0394 be the graph Laplacian, and suppose that h : V \u2192 R is a function satisfying the coercive condition on \u03a9, namely there exists some constant \u03b4 \u203a 0 such that $$\u222b_\u03a9u(-\u0394+h)ud\u03bc \u2265 \u03b4 \u222b_\u03a9|\u2207u|\u00b2d\u03bc,\\quad \u2200u:V \u2192 R.$$ By the mountain-pass theoremof Ambrosette-Rabinowitz, we prove that for any p \u203a 2, there exists a positive solution to $$-\u0394u+hu=|u|^{p-2}u\\quad\\;\\; in\\;\\; \u03a9$$. Using the same method, we prove similar results for the p-Laplacian equations. This partly improves recent results of Grigor'yan-Lin-Yang."