Let $p$ be a prime number with $p \equiv 3\,\mathrm{mod}\,4$, and let $k=\mathbf{Q}(\sqrt{-p})$. Denote by $h_{n}^{-}$ the relative class number of the $n$th layer of the cyclotomic $\mathbf{Z}_{p}$-extension over $k$. Let $q=(p-1)/2$ and $d_{p}$ be the largest divisor of $q$ with $d_{p} < q$. Let $\ell$ be a prime number with $\ell \neq p$. We show that $\ell \nmid h_{n}^{-}$ for all $n \geq 0$ if $\ell \geq q-2d_{p}$ and $\ell$ is a primitive root modulo $p^{2}$.

In this paper, we use the variational methods to study the following problem \begin{equation} {-}Δ_{\mathbf{B}^{N}}u=(d(x))^{α}|u|^{p-2}u, u\in H_{r}^{1}(\mathbf{B}^{N}) \label{Lb1} \end{equation} in Hyperbolic space $\mathbf{B}^{N}$, where $\alpha>0$, $d(x)=d_{\mathbf{B}^{N}}(0,x)$, and $H_{r}^{1}(\mathbf{B}^{N})$ denote the Sobolev space of radial $H^{1}(\mathbf{B} ^{N})$ function on the disc model of the Hyperbolic space $\mathbf{B}^{N}$ and $\Delta_{\mathbf{B}^{N}}$ denotes the Laplace-Beltrami operator on $\mathbf{B}^{N}$, $N\geq 3$. Unlike the corresponding problem in Euclidean space $\mathbf{R}^{N}$, we prove that there exists a positive solution of problem (1) provided that $p\in (2, \frac{2N+2\alpha}{N-2})$ which will be contrasted with a result due to Gidas and Spruck [6].

In this article, we investigate some properties of some difference polynomials. The results in this article improve some theorems of Liu and Laine. Several examples are provided to show that our results are best possible.

We study the growth of solutions of a certain difference equations, and study the uniqueness question of entire functions of finite orders sharing an entire function of smaller order with their shifts. The uniqueness results in this paper also extend and improve Theorem 1 [11].