A Capacitary Approach to Semilinear Elliptic Inequalities with Potentials on Weighted Graphs

Mathematics > Analysis of PDEs [Submitted on 17 Jun 2026] Title:A Capacitary Approach to Semilinear Elliptic Inequalities with Potentials on Weighted Graphs View PDF HTML (experimental)Abstract:We develop a capacitary approach to semilinear elliptic inequalities on weighted graphs with a potential. More precisely, we study the nonexistence of nontrivial nonnegative solutions of \[ \Delta u+w(x)u+v(x)u^\sigma\le0 \qquad\text{in }V, \] where \((V,\omega,\mu)\) is a connected, locally finite weighted graph, \(\Delta\) is the associated graph Laplacian, \(\sigma>1\), \(v>0\), and \(w\) is a real-valued potential. The potential term is handled by means of a positive solution \(H\) of \(\Delta H+wH=0\), which transforms the operator \(\Delta+w\) into the \(H\)-Laplacian associated with a new weighted graph. Our main nonexistence criterion is formulated directly in terms of cut-off functions and the regions where their \(H\)-Laplacian is controlled. Unlike metric criteria based on pseudo-metric annuli, our formulation determines the capacitary sets from the support of the \(H\)-Laplacian estimates for the cut-off functions. We provide an example showing that our result applies in situations not covered by previous nonexistence criteria based on structural lower bounds or pseudo-metric annular volume estimates. We also show that the growth exponent in our capacitary condition is sharp by constructing an example for which the condition fails by an arbitrary power \(R^\varepsilon\), while a positive nontrivial solution exists. References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.