arXiv subjects

Zhi-Yun Tang

Publications and source records attributed to Zhi-Yun Tang.

4 recordsLinked to original sources

arXiv2026

Multiplicity and asymptotics of positive solutions for critical-concave Kirchhoff equation

This paper focuses on the critical Kirchhoff equation with concave perturbation \begin{align*} \begin{cases} \displaystyle -\Big(a+b\int_Ω|\nabla u|^2dx\Big)Δu=|u|^4u+λ|u|^{q-2}u\ \ &\mbox{in}\ Ω, \displaystyle u=0\ \ &\mbox{on}\ \partialΩ, \end{cases} \end{align*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^3$, $a,b,λ>0$ and $1 0$ small enough. Moreover, we analyse the asymptotic behaviour of positive solutions as $b\rightarrow0$ and $λ\rightarrow0$, respectively. This work is a counterpart of [A. Ambrosetti et al., J.~Funct.~Anal. 1994] for the Kirchhoff equation. It is noteworthy that we don't require that $b>0$ is small enough here, which is imposed in the existing literatures to make refined estimates for the mountain pass level.

math.AP↗

arXiv2025

The Li-Lin's open problem on $\mathbb{R}^N$

In 2012, Y.Y. Li and C.-S. Lin (Arch. Ration. Mech. Anal., 203(3): 943-968) posed an open problem concerning the existence of positive solutions to the elliptic equation $$ \begin{cases} -Δu = -λ|x|^{-s_1}|u|^{p-2}u + |x|^{-s_2}|u|^{q-2}u & \text{in } Ω, u = 0 & \text{on } \partial Ω, \end{cases} $$ for $λ> 0$, $p > q = 2^*(s_2)$, $0 \leq s_1 < s_2 < 2$, and $2^*(s) = \frac{2(N-s)}{N-2}$ denotes the Hardy-Sobolev critical exponent, initially studied in bounded domains $Ω\subset \mathbb{R}^N$, $N \geq 3$. Currently, research on this open problem remains limited, and a complete resolution is still far from being achieved. Motivated by the need to address this open problem in more general settings, we extend our investigation to the entire space $\mathbb{R}^N$, focusing on the equation $$ -Δu + u = -λ|x|^{-s_1}|u|^{p-2}u + |x|^{-s_2}|u|^{q-2}u \quad \text{in } \mathbb{R}^N. $$ Our analysis reveals stark contrasts between bounded and unbounded domains: in $\mathbb{R}^N$, the equation admits no solution when $q = 2^*(s_2)$ for any $λ> 0$, whereas a positive solution exists when $q < 2^*(s_2)$. To establish these results, we employ the Nehari manifold method; however, the functional's unboundedness from below on the manifold causes standard global minimization techniques to be inapplicable. Instead, we characterize a local minimizer of the energy functional on the Nehari manifold, overcoming the challenge posed by the lack of a global minimizer.

math.AP↗

arXiv2025

A positive solution of the elliptic equation on a starshaped domain with boundary singularities

We consider the elliptic equation with boundary singularities \begin{equation} \begin{cases} -Δu=-λ|x|^{-s_{1}}|u|^{p-2}u+|x|^{-s_{2}}|u|^{q-2}u &\text { in } \varOmega , u(x)=0 &\text { on } \partial \varOmega , \end{cases} \end{equation} where $0\leq s_1 < s_2 < 2$, $2 q>\frac{2-s_2}{2-s_1}p+\frac{2s_2-2s_1}{2-s_1}$. We also discuss the asymptotic behavior of the positive solution and find a new class of blow-up points by blowing up analysis. These blow-up points are on the boundary of the domain, which are not similar with the usual.

math.AP↗

arXiv2024

On Li-Lin's open problem

In this paper, we give a first negative answer to a question proposed by Li and Lin (Arch Ration Mech Anal 203(3): 943-968, 2012). Meanwhile we also give a second positive answer to the Li-Lin's open problem. The first positive answer was given by G. Cerami, X. Zhong and W. Zou (Calc. Var. Partial Differential Equations, 54(2): 1793-1829, 2015).

math.AP↗