Panchromatic 3-colorings of random hypergraphs

The paper deals with panchromatic 3-colorings of random hypergraphs. A vertex 3-coloring is said to be panchromatic for a hypergraph if every color can be found on every edge. Let $H(n,k,p)$ denote the binomial model of a random $k$-uniform hypergraph on $n$ vertices. For given fixed $c>0$, $k⩾3$ and $p=cn∕\left(\genfrac{}{}{0ex}{}{n}{k}\right)$, we prove that if $c<\frac{ln3}{3}\cdot {\left(\frac{3}{2}\right)}^k-\frac{ln3}{2}-O\left({\left(\frac{\sqrt{3}}{2}\right)}^k\right)$then $H(n,k,p)$ admits a panchromatic 3-coloring with probability tending to 1 as $n\to \infty$, but if $k$ is large enough and $c>\frac{ln3}{3}\cdot {\left(\frac{3}{2}\right)}^k-\frac{ln3}{2}+O\left({\left(\frac{3}{4}\right)}^k\right)$then $H(n,k,p)$ does not admit a panchromatic 3-coloring with probability tending to 1 as $n\to \infty$.     

Infinitely many solutions to a quasilinear Schrödinger equation with a local sublinear term

This paper discusses the quasilinear Schrödinger equation $-\Delta u+V\left(x\right)u-\Delta \left[{(1+u^2)}^{\frac{1}{2}}\right]\frac{u}{2{(1+u^2)}^{\frac{1}{2}}}=K\left(x\right)f\left(u\right),x\in {\mathbb{R}}^N,$where $N⩾3$. Under appropriate assumptions on the potentials $V$ and $K$ and local sublinear growth assumptions on the nonlinear term $f$, we get the existence of infinitely many nontrivial solutions by using a revised Clark theorem and a priori estimate of the solution.