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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 $.
In this paper we state some sharp maximum principle, i.e. we characterize the geometry of the sets of minima for supersolutions of equations involving the $k$
-th fractional truncated Laplacian or the $k$
-th fractional eigenvalue which are fully nonlinear integral operators whose nonlocality is somehow $k$
-dimensional.
In this paper we show the existence of solution for the following class of semipositone problem
P
where N ≥ 3, a > 0, h : ℝN → (0, + ∞) and f : [0, + ∞) → [0, + ∞) are continuous functions with f having a subcritical growth. The main tool used is the variational method together with estimates that involve the Riesz potential.
$$\left\{\matrix{-\Delta u & = & h(x)(f(u)-a) & \hbox{in} & {\open R}^N, \cr u & \gt & 0 & \hbox{in} & {\open R}^N, \cr}\right.$$
