Recently, the discrete reversed Hardy–Littlewood–Sobolev inequality with infinite terms was proved. In this article, we study the attainability of its best constant. For this purpose, we introduce a discrete reversed Hardy–Littlewood–Sobolev inequality with finite terms. The constraint of parameters of this inequality is more relaxed than that of parameters of inequality with infinite terms. We here show the limit relations between their best constants and between their extremal sequences. Based on these results, we obtain the attainability of the best constant of the inequality with infinite terms in the noncritical case.

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 are concerned with the non-existence of positive solutions of a Hartree–Poisson system:

\begin{equation*}\left\{\begin{aligned}&-\Delta u=\left(\frac{1}{|x|^{n-2}}\ast v^p\right)v^{p-1},\quad u \gt 0\ \text{in} \ \mathbb{R}^{n},\\&-\Delta v=\left(\frac{1}{|x|^{n-2}}\ast u^q\right)u^{q-1},\quad v \gt 0\ \text{in} \ \mathbb{R}^{n},\end{aligned}\right.\end{equation*}

$n \geq3$

$\min\{p,q\} \gt 1$

We prove a reversed Hardy–Littlewood–Pólya inequality with finite terms. We also give the limit of the best constant.

This paper is concerned with the p-Ginzburg–Landau (p-GL) type model with $p\neq 2$. First, we obtain global energy estimates and energy concentration properties by the singularity analysis. Next, we give a decay rate of $1-|u_\varepsilon |$ in the domain away from the singularities when $\varepsilon \to 0$, where $u_\varepsilon $ is a minimizer of p-GL functional with $p \in (1,2)$. Finally, we obtain a Liouville theorem for the finite energy solutions of the p-GL equation on $\mathbb {R}^2$.

In this paper, we study negative classical solutions and stable solutions of the following k-Hessian equation

$$F_k(D^2V) = (-V)^p\quad {\rm in}\;\; R^n$$