arXiv subjects

Yunyun Hu

Publications and source records attributed to Yunyun Hu.

5 recordsLinked to original sources

arXiv2026

Classification of solutions to the Liouville equation with a nonlinear Robin boundary condition on the unit disk

In this paper, we study the nonlinear boundary value problem \begin{equation*} \begin{cases} -Δu=e^{2u},& \mbox{in } {\mathbb{D}},\\ \frac{\partial u}{\partialν}+λ=e^u ,& \mbox{on } {\mathbb{S}^{1}}, \end{cases} \end{equation*} where $\mathbb D$ is the unit disk, $λ$ is a constant and $ν$ denotes the outer unit normal on $\mathbb S^1$. For $0<λ\le2$, we establish a complete classification of smooth solutions. For $2<λ\leq3$, we prove a dichotomy between radial solutions and nonradial solutions. We develop a Hardy-Wronskian boundary rigidity method that transforms the nonlinear boundary value problem into a spectral rigidity problem for normalized holomorphic frames. This method exploits the complex-analytic structure of the Liouville equation with a Robin boundary condition and provides a new rigidity framework for related two-dimensional elliptic boundary value problems.

math.AP↗

arXiv2023

Reversed Hardy-Littlewood-Sobolev inequalities with vertical weights on the upper half space

In this paper, we obtain the reversed Hardy-Littlewood-Sobolev inequality with vertical weights on the upper half space and discuss the extremal functions. We show that the sharp constants in this inequality are attained by introducing a renormalization method. The classification of corresponding extremal functions is discussed via the method of moving spheres. Moreover, we prove the sufficient and necessary conditions of existence for positive solutions to the Euler-Lagrange equations by using Pohozaev identities in weak sense. This renormalization method is rearrangement free, which can be also applied to prove the existence of extremal functions for sharp (reversed) Hardy-Littlewood-Sobolev inequality with extended kernels and other similar inequalities.

math.AP↗

arXiv2020

Asymptotic method of moving planes for fractional parabolic equations

In this paper, we develop a systematical approach in applying an asymptotic method of moving planes to investigate qualitative properties of positive solutions for fractional parabolic equations. We first obtain a series of needed key ingredients such as narrow region principles, and various asymptotic maximum and strong maximum principles for antisymmetric functions in both bounded and unbounded domains. Then we illustrate how this new method can be employed to obtain asymptotic radial symmetry and monotonicity of positive solutions in a unit ball and on the whole space. Namely, we show that no matter what the initial data are, the solutions will eventually approach to radially symmetric functions. We firmly believe that the ideas and methods introduced here can be conveniently applied to study a variety of nonlocal parabolic problems with more general operators and more general nonlinearities.

math.AP↗

arXiv2020

Sharp reversed Hardy-Littlewood-Sobolev inequality with extended kernel

In this paper, we prove the following reversed Hardy-Littlewood-Sobolev inequality with extended kernel \begin{equation*} \int_{\mathbb{R}_+^n}\int_{\partial\mathbb{R}^n_+} \frac{x_n^β}{|x-y|^{n-α}}f(y)g(x) dydx\geq C_{n,α,β,p}\|f\|_{L^{p}(\partial\mathbb{R}_+^n)} \|g\|_{L^{q'}(\mathbb{R}_+^n)} \end{equation*} for any nonnegative functions $f\in L^{p}(\partial\mathbb{R}_+^n)$ and $g\in L^{q'}(\mathbb{R}_+^n)$, where $n\geq2$, $p,\ q'\in (0,1)$, $α>n$, $0\leqβ<\frac{α-n}{n-1}$, $p>\frac{n-1}{α-1-(n-1)β}$ such that $\frac{n-1}{n}\frac{1}{p}+\frac{1}{q'}-\frac{α+β-1}{n}=1$. We prove the existence of extremal functions for the above inequality. Moreover, in the conformal invariant case, we classify all the extremal functions and hence derive the best constant via a variant method of moving spheres, which can be carried out \emph{without lifting the regularity of Lebesgue measurable solutions}. Finally, we derive the sufficient and necessary conditions for existence of positive solutions to the Euler-Lagrange equations by using Pohozaev identities. Our results are inspired by Hang, Wang and Yan \cite{HWY}, Dou, Guo and Zhu \cite{DGZ} for $α<n$ and $β=1$, and Gluck \cite{Gl} for $α<n$ and $β\geq0$.

math.AP↗

arXiv2016

Two-bubble nodal solutions for slightly subcritical Fractional Laplacian

In this paper, we consider the existence of nodal solutions with two bubbles to the slightly subcritical problem with the fractional Laplacian \begin{equation*} \left\{\aligned &(-Δ)^su=|u|^{p-1-\varepsilon}u\ \ \mbox{in}\ Ω&u=0\ \mbox{on}\ \partialΩ, \endaligned \right. \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb R^N$, $N>2s$, $0 0$ is a small parameter, which can be seen as a nonlocal analog of the results of Bartsch, Micheletti and Pistoia (2006) \cite{Bartsch1}.

math.AP↗