• Abstract

In the paper, the authors introduce the Orlicz space corresponding to the Young function and, by virtue of the equivalent theorem between the modified $K$-functional and modulus of smoothness, establish the direct, inverse, and equivalent theorems for the linear combinations of Jacobi weighted Baskakov--Durrmeyer operators in the Orlicz spaces.

• General Summary

• Abstract

This paper obtains a criterion on the weighted $L^{p}$-boundedness for a class of multilinear oscillatory singular operators with real-valued polynomial phases and rough kernels belonging to the block spaces.

• General Summary

• Abstract

In this paper, we systematically study weak solutions of a linear singular or degenerate parabolic equation in a mixed divergence form and nondivergence form, which arises from the linearized fast diffusion equation and the linearized porous medium equation with the homogeneous Dirichlet boundary condition. We prove the Holder regularity of their weak solutions.

• General Summary

• Abstract

In this paper, we study the existence of solutions for Kirchhoff equation

with mass constraint condition

where $a$, $b$, $c>0$, $\mu\in \mathbb{R}$ and $2<q<p<6$. The $\lambda \in \mathbb{R}$ appears as a Lagrange multiplier. For the range of $p$ and $q$, the Sobolev critical exponent $6$ and mass critical exponent $\frac{14}{3}$ are involved which corresponding energy functional is unbounded from below on $S_{c}$. We consider the defocusing case, i.e. $\mu<0$ when $(p, q)$ belongs to a certain domain in $\mathbb{R}^{2}$. We prove the existence and multiplicity of normalized solutions by using constraint minimization, concentration compactness principle and Minimax methods. We partially extend the results have been studied.

• General Summary