In this paper, we consider the existence of multiple solutions following critical nonlocal elliptic equations with magnetic field: \begin{document}$\begin{align} \left\{\begin{aligned} (-i\nabla-A(x))^2u& = \lambda |u|^{p-2}u+\left(\int_{\Omega}\frac{|u(y)|^{2^*_\alpha}}{|x-y|^{\alpha}}dy\right)|u|^{2^*_\alpha-2}u\quad {\rm in}\quad \Omega,\\ &u 0\quad \partial\Omega,\\ \end{aligned}\ri...

We obtain asymptotic for the quantity $\int_0^1 \bigg|\sum_{n\le X}\tau_k(n)e(n\alpha)\bigg|d\alpha$ where $\tau_k(n) = \sum_{d_1\dots d_k n} 1$. This follows from a quick application of circle method. Along way, we find minor arc bounds exponential sum with $\tau_k$, and asymptotics high moments Dirichlet kernel.

<p style='text-indent:20px;'>In this paper, we consider the weighted fourth order equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \Delta(|x|^{-\alpha}\Delta u)+\lambda \text{div}(|x|^{-\alpha-2}\nabla u)+\mu|x|^{-\alpha-4}u = |x|^\beta u^p\quad \text{in} \quad \mathbb{R}^n \backslash \{0\}, $\end{document...

Abstract We study the minimal gap statistic for sequences of form $\left( \alpha x_n \right)_{n = 1}^{\infty}$ where is a sequence real numbers, and its connection to additive energy 1}^{\infty}$. Inspired by recent paper Aistleitner, El-Baz Munsch we show conditionally on Lindelöf Hypothesis that if lowest possible order then almost all α, $\delta_{\min}^{\alpha} (N) \min \left\{\alpha x_m - \...

Let $$X^N$$ be a family of $$N\times N$$ independent GUE random matrices, $$Z^N$$ deterministic P self-adjoint noncommutative polynomial, that is for any N, $$P(X^N,Z^N)$$ self-adjoint, f smooth function. We prove k, if enough, there exist constants $$\alpha _i^P(f,Z^N)$$ such $$\begin{aligned} \mathbb {E}\left[ \frac{1}{N}\text {Tr}\left( f(P(X^N,Z^N)) \right) \right] \ =\ \sum _{i=0}^k \frac{...

In this paper, we prove the following reversed Hardy–Littlewood–Sobolev inequality with extension kernel: $$\int _{\mathbb R_+^n}\int _{\partial\mathbb R^n_+}\frac{x_n^\beta}{|x-y|^{n-\alpha}}f(y)g(x)\,dy\,dx\geq C_{n,\alpha ,\beta ,p}\|f\|_{L^{p}(\partia

We establish sufficient conditions for the existence and uniqueness of $$(N,\lambda )$$ -periodic solutions following abstract model: $$\begin{aligned} \Delta ^{\alpha }u(n)=Au(n+1)+f(n,u(n)), \quad n\in {\mathbb {Z}}, \end{aligned}$$ where $$0 < \alpha \le 1 $$ , A is a closed linear operator defined in Banach space X, $$\Delta }$$ denotes fractional difference Weyl-like sense, f satisfies app...

In this paper, our purpose is to study the following weighted integral equation with negative power$ \begin{equation} f^{q-1}(x) = \int_{\Omega}\dfrac {|x|^{\alpha}|y|^{\alpha}f(y)}{|x-y|^{N-\mu}} \ dy +\lambda\int_{\Omega}\dfrac {f(y)}{|x-y|^{N-\mu-1}} , f\geq0, x\in\overline{\Omega}\;(0.1) \end{equation} $where $ \Omega a bounded domain smooth boundary in \mathbb{R}^N $, 0\in \mu&gt;N \alpha&...