• Abstract

In this paper we study the following nonlinear Schrodinger system:

Here, $a,\,b,\,\lambda\in C(\mathbb R^N,\,\mathbb R)$ are all non-periodic in $x_i$ for $i=1,\,\cdots,\,N$, $N\geq 3$,  $f,\,g\in C(\mathbb R^2,\,\mathbb R)$. We show that this system has infinitely many solutions with small negative energies  and infinitely many  large-energy  solutions. To the best of our knowledge, there is no corresponding result about such a  Schrodinger system.

• General Summary

• Abstract

Let $(\mathcal{X},d,\mu)$ be a non-homogeneous metric measure space satisfying both the upper doubling and the geometrically doubling conditions.  In this paper, we study weighted inequalities of the Calderon-Zygmund operator on $(\mathcal{X},d,\mu)$. Specifically, for $1 <p< \infty$, we identify sufficient conditions for the weight on one side, which guarantee the existence of another weight in the other side, so that the weighted $L^p$ inequality holds.  We deal with this problem by developing a vector-valued theory for Calderon-Zygmund operators on the non-homogeneous metric measure spaces which is interesting in its own right.

• General Summary

• Abstract

 This paper study a general nonlocal problem characterized by the equation:

$$-\mathcal{L}_K u+V(x)u=P(x)h(u)\quad \text{in} \ \ \mathbb{R}.$$

Here,  $\mathcal{L}_K$ represents a nonlocal integro-differential operator, and $h$ is a nonlinear term displaying subcritical  or critical growth of Trudinger-Moser type. We initially prove the existence of a ground state solution by employing variational  methods and innovative analytical approaches. Furthermore, through the application of constrained variational methods, Brouwer  degree theory, and a quantitative deformation lemma, we establish the existence of a sign-changing solution with minimal energy,  surpassing the energy of the ground state solution.

• General Summary

• Abstract

We aim to present a innovation extension of the mean-square stochastic integral of a process  on time scale based on a $\mathfrak{q}$-operator and fractional $\mathfrak{q}$-operator. By presenting the $\mathfrak{q}$-convex definition for a stochastic process and using quantum operators, we were able to prove some well-known inequalities such as the Hermit-Hadamardand the Jensen for the processes.

• General Summary

• Abstract

For complex-valued functions $f\in L^1(\mathbb R^2_+)$,  where $\mathbb R_+:= [0,\infty)$ we give sufficient conditions under which the double cosine or cosine-sine Fourier transform of $f$ belongs to a generalized Lipschitz class defined by the mixed modulus of smoothness of orders $m,n\in\mathbb N=\{1,2,\cdots\}$ in uniform metric. The sharpness of these conditions is established  under some restriction for non-negative functions.

• General Summary