Abstract
Abstract
We use potential analysis to study the properties of positive solutions of a discrete Wolff-type equation
$$ \begin{align*} w(i)=W_{\beta,\gamma}(w^q)(i), \quad i \in \mathbb{Z}^n. \end{align*} $$
Here,
$n \geq 1$
,
$\min \{q,\beta \}>0$
,
$1<\gamma \leq 2$
and
$\beta \gamma <n$
. Such an equation can be used to study nonlinear problems on graphs appearing in the study of crystal lattices, neural networks and other discrete models. We use the method of regularity lifting to obtain an optimal summability of positive solutions of the equation. From this result, we obtain the decay rate of
$w(i)$
when
$|i| \to \infty $
.
Publisher
Cambridge University Press (CUP)