Liouville-type theorems for a nonlinear fractional Choquard equation

In this paper, we are concerned with the fractional Choquard equation on the whole space ${\mathbb{R}}^N$ $${(-\mathrm{\Delta })}^{s}u=\left(\frac{1}{{|x|}^{N-2s}}\ast u^p\right)u^{p-1}$$ with , $N>2s$ and $p\in \mathbb{R}$. We first prove that the equation does not possess any positive solution for $p\le 1$. When $p>1$, we establish a Liouville type theorem saying that if then the equation has no positive stable solution. This extends, in particular, a result in [27] to the fractional Choquard equation.     

Study on the Isomorphous Substitution of Silicon by Indium in the MFI Framework

In-ZSM-5 zeolites were prepared by hydrothermal synthesis and characterized by XRD, MAS NMR, IR, thermoanalysis, TPD of ammonia and catalytically tested. The data reveal the isomorphous substitution of silicon by indium in the framework, however, in comparison with other elements the substitution degree is low: Al > Ga > Fe > B > In. In-ZSM-5 shows a low catalytic activity.