arXiv subjects

Shaolong Peng

Publications and source records attributed to Shaolong Peng.

5 recordsLinked to original sources

arXiv2026

Rigidity of weak solutions for anisotropic N-Laplacian equation with Neumann or Robin boundary condition

This paper is devoted to the rigidity of weak solutions for anisotropic $N$-Laplacian equations with Neumann or Robin boundary conditions on smooth bounded convex domains of $\mathbb{R}^N$. The anisotropic operator is given by $$a(\xi) = H^{N-1}(\xi)\nabla H(\xi),$$ where $H$ stands for a norm on $\mathbb{R}^N$; this formulation contains the classical $N$-Laplacian as a special case. We establish a key integral inequality involving the anisotropic gradient and the second fundamental form of the domain boundary, which acts as the core technical tool in our proofs. Under natural monotonicity assumptions on the nonlinearity, we prove that all weak solutions to the Neumann boundary problem are constant, without requiring any a priori boundedness assumption on the solution. Furthermore, we extend this rigidity result to Robin boundary value problems by imposing suitable constraints on the boundary nonlinear term. Moreover, our rigidity results remain valid not only on bounded convex domains but also on suitable unbounded domains. By working under substantially weaker assumptions than those previously available, we establish rigidity results that fill the gaps in the existing literature for anisotropic $N$-Laplacian equations with nonlinear boundary conditions and substantially extend the rigidity theory of anisotropic quasilinear elliptic equations at the critical exponent $p=N$.

math.AP↗

arXiv2026

Nonlinear Neumann boundary problems for $n$-Laplacian Liouville equation on a half space

In this paper, for general $n\geq2$, we classify solutions to $n$-Laplacian Liouville equation with positive nonlinear Neumann boundary condition on the half-space $\mathbb{R}^{n}_{+}$. Under the positive nonlinear Neumann boundary condition, our result extend the classification result for the second order Liouville equation in \cite{Li} from $n=2$ to general $n\geq2$, and also extend the classification result for critical $p$-Laplacian equation in \cite{Zhou} from $p<n$ to $p=n$.

math.AP↗

arXiv2025

On the sharp quantitative stability of critical points of the Hardy-Littlewood-Sobolev inequality in $\mathbb{R}^{n}$ with $n\geq3$

Assume $n\geq3$ and $u\in \dot{H}^1(\mathbb{R}^n)$. Recently, Piccione, Yang and Zhao \cite{Piccione-Yang-Zhao} established a nonlocal version of Struwe's decomposition in \cite{Struwe-1984}, i.e., if $\Gamma(u):=\left\|\Delta u+D_{n,\alpha}\int_{\mathbb{R}^{n}}\frac{|u|^{p_{\alpha}}(y) }{|x-y|^{\alpha}}\mathrm{d}y |u|^{p_{\alpha}-2} u\right\|_{\dot{H}^{-1}} \rightarrow 0$ and $u\geq 0$, then $dist(u,\mathcal{T})\to 0$, where $dist(u,\mathcal{T})$ denotes the $\dot{H}^1(\mathbb{R}^n)$-distance of $u$ from the manifold of sums of Talenti bubbles. In this paper, we establish the nonlocal version of the quantitative estimates of Struwe's decomposition in Ciraolo, Figalli and Maggi \cite{CFM} for one bubble and $n\geq3$, Figalli and Glaudo \cite{Figalli-Glaudo2020} for $3\leq n\leq5$ and Deng, Sun and Wei \cite{DSW} for $n\geq6$ and two or more bubbles. We prove that for $n\geq 3$ and $0<\alpha<n$, \[dist (u,\mathcal{T})\leq C\begin{cases} \Gamma(u)\left|\log \Gamma(u)\right|^{\frac{1}{2}}\quad&\text{if } \,\, n\geq 6, \,\, \nu\geq2 \,\, \text{and} \,\, \alpha=\frac{n+2}{2}, \\ \Gamma(u) \quad&\text{for any other cases,}\end{cases}\] where $\nu$ denotes the number of bubbles. Furthermore, we show that this inequality is sharp for $n\geq 6$ and $\alpha=\frac{n+2}{2}$. It should be emphasized that, in our paper, we have developed new techniques to deal with the strong singular case $4<\alpha<n$, which can not be handled by reduction methods in previous works. We believe that our method can also be applied to other problems related to the physically interesting Hartree equation.

math.AP↗

arXiv2022

Non-degeneracy of solution for critical Lane-Emden systems with linear perturbation

In this paper, we consider the following elliptic system \begin{equation*} \begin{cases} -\Delta u = |v|^{p-1}v +\epsilon(\alpha u + \beta_1 v), &\hbox{ in }\Omega, \\-\Delta v = |u|^{q-1}u+\epsilon(\beta_2 u +\alpha v), &\hbox{ in }\Omega, \\u=v=0,&\hbox{ on }\partial\Omega, \end{cases} \end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$, $N\geq 3$, $\epsilon$ is a small parameter, $\alpha$, $ \beta_1$ and $ \beta_2$ are real numbers, $(p,q)$ is a pair of positive numbers lying on the critical hyperbola \begin{equation*} \begin{split} \frac{1}{p+1}+\frac{1}{q+1} =\frac{N-2}{N}. \end{split} \end{equation*} We first revisited the blowing-up solutions constructed in \cite{Kim-Pis} and then we proved its non-degeneracy. We believe that the various new ideas and technique computations that we used in this paper would be very useful to deal with other related problems involving critical Halmitonian system and the construction of new solutions.

math.AP↗

arXiv2018

Liouville type theorems, a priori estimates and existence of solutions for non-critical higher order Lane-Emden-Hardy equations

In this paper, we are concerned with the non-critical higher order Lane-Emden-Hardy equations \begin{equation*} (-Δ)^{m}u(x)=\frac{u^{p}(x)}{|x|^{a}} \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n} \end{equation*} with $n\geq3$, $1\leq m<\frac{n}{2}$, $0\leq a<2m$, $1<p<\frac{n+2m-2a}{n-2m}$ if $0\leq a<2$, and $1<p<\infty$ if $2\leq a<2m$. We prove Liouville theorems for nonnegative classical solutions to the above Lane-Emden-Hardy equations (Theorem \ref{Thm0}), that is, the unique nonnegative solution is $u\equiv0$. As an application, we derive a priori estimates and existence of positive solutions to non-critical higher order Lane-Emden equations in bounded domains (Theorem \ref{Thm1} and \ref{Thm2}). The results for critical order Hardy-Hénon equations have been established by Chen, Dai and Qin \cite{CDQ} recently.

math.AP↗