Conjecture on critical and subcritical solutions for discrete p-biharmonic equations and Lane–Emden systems

Let $G$ be a graph of polynomial growth with homogeneous dimension $N\geq 3$. For the discrete $p$-biharmonic equation

$$\Delta\left(\lvert\Delta u\rvert^{p-2}\Delta u\right)-\lvert u\rvert^{q-2}u=0,$$

where $1<p<N/2$, and for the discrete Lane–Emden system

$$\begin{cases} -\Delta u=\lvert v\rvert^{p^{\prime}-2}v,\\ -\Delta v=\lvert u\rvert^{q-2}u, \end{cases}$$

with $p,q>1$, $p^{\prime}=p/(p-1)$, $q^{\prime}=q/(q-1)$, and $u\in D^{2,p}(G)$, $v\in D^{2,q^{\prime}}(G)$, call the cases satisfying $1/p^{\prime}+1/q=(N-2)/N$ critical and those satisfying $1/p^{\prime}+1/q<(N-2)/N$ subcritical. Critical and subcritical solution conjecture. The discrete $p$-biharmonic equation and the discrete Lane–Emden system have positive solutions in the critical cases, whereas every non-negative solution in the subcritical cases is trivial. The conjecture extends corresponding continuous results to graphs of polynomial growth; the paper establishes positive solutions in supercritical cases, while the critical and subcritical cases remain unresolved here.