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

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 $$\mathrm{\Delta }u+w(x)u+v(x)u^{\sigma }\le 0\text{in }V,$$ where $(V,\omega ,\mu )$ is a connected, locally finite weighted graph, $\mathrm{\Delta }$ 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 $\mathrm{\Delta }H+wH=0$, which transforms the operator $\mathrm{\Delta }+w$ 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 $R^{\epsilon }$, while a positive nontrivial solution exists.