arXiv subjects

Gongbao Li

Publications and source records attributed to Gongbao Li.

6 recordsLinked to original sources

arXiv2022

Further Estimates with Pseudogamma functions

The pseudo-Gamma function is a key tool introduced recently by Cheng and Albeverio in the proof of \break the density hypothesis. This function is doubly symmetric, which means that it is reflectively symmetric about the real axis by the Schwarz principle, whereas it is also reflectively symmmetric about the half line where the real part of the variable is equal to $\tfrac{1}{2}$. In this article, we sharpen the estimate given in the proof of the density hypothesis for this doubly symmetric pseudo-Gamma function on the real axis near the symmetry center by taking a different approach from the way used in the density hypothesis proof directly from the definition, reducing the error caused by the fact that the difference of two pivotal parameters in the definition of the pseudo-Gamma function is much larger than the difference of the variables in this particular case.

math.NT↗

arXiv2021

Normalized solutions to a class of Kirchhoff equations with Sobolev critical exponent

In this paper, we consider the existence and asymptotic properties of solutions to the following Kirchhoff equation \begin{equation}\label{1}\nonumber - \Bigl(a+b\int_{\R^3} {{{\left| {\nabla u} \right|}^2}}\Bigl) Δu =λu+ {| u |^{p - 2}}u+μ{| u |^{q - 2}}u \text { in } \mathbb{R}^{3} \end{equation} under the normalized constraint $\int_{{\mathbb{R}^3}} {{u}^2}=c^2$, where $a\!>\!0$, $b\!>\!0$, $c\!>\!0$, $2\!<\!q\!<\!\frac{14}{3}\!<\! p\!\leq\!6$ or $\frac{14}{3}\!<\!q\!<\! p\!\leq\! 6$, $μ\!>\!0$ and $λ\!\in\!\R$ appears as a Lagrange multiplier. In both cases for the range of $p$ and $q$, the Sobolev critical exponent $p\!=\!6$ is involved and the corresponding energy functional is unbounded from below on $S_c=\Big\{ u \in H^{1}({\mathbb{R}^3}): \int_{{\mathbb{R}^3}} {{u}^2}=c^2 \Big\}$. If $2\!<\!q\!<\!\frac{10}{3}$ and $\frac{14}{3}\!<\! p\!<\!6$, we obtain a multiplicity result to the equation. If $2\!<\!q\!<\!\frac{10}{3}\!<\! p\!=\!6$ or $\frac{14}{3}\!<\!q\!<\! p\!\leq\! 6$, we get a ground state solution to the equation. Furthermore, we derive several asymptotic results on the obtained normalized solutions. Our results extend the results of N. Soave (J. Differential Equations 2020 $\&$ J. Funct. Anal. 2020), which studied the nonlinear Schrödinger equations with combined nonlinearities, to the Kirchhoff equations. To deal with the special difficulties created by the nonlocal term $({\int_{\R^3} {\left| {\nabla u} \right|} ^2}) Δu$ appearing in Kirchhoff type equations, we develop a perturbed Pohozaev constraint approach and we find a way to get a clear picture of the profile of the fiber map via careful analysis. In the meantime, we need some subtle energy estimates under the $L^2$-constraint to recover compactness in the Sobolev critical case.

math.AP↗

arXiv2019

The existence of a nontrivial weak solution to a double critical problem involving fractional Laplacian in ${\R}^n$ with a Hardy term

In this paper, we consider the existence of nontrivial weak solutions to a double critical problem involving fractional Laplacian with a Hardy term: \begin{equation} \label{eq0.1} (-Δ)^{s}u-γ {\frac{u}{|x|^{2s}}}= {\frac{{|u|}^{ {2^{*}_{s}}(β)-2}u}{|x|^β}}+ \big [ I_μ* F_α(\cdot,u) \big](x)f_α(x,u), \ \ u \in {\dot{H}}^s(\R^{n}) \end{equation} where $s \in(0,1)$, $0\leq α,β<2s<n$, $μ\in (0,n)$, $γ<γ_{H}$, $I_μ(x)=|x|^{-μ}$, $F_α(x,u)=\frac{ {|u(x)|}^{ {2^{\#}_μ }(α)} }{ {|x|}^{ {δ_μ (α)} } }$, $f_α(x,u)=\frac{ {|u(x)|}^{{ 2^{\#}_μ }(α)-2}u(x) }{ {|x|}^{ {δ_μ (α)} } }$, $2^{\#}_μ (α)=(1-\fracμ{2n})\cdot 2^{*}_{s} (α)$, $δ_μ (α)=(1-\fracμ{2n})α$, ${2^{*}_{s}}(α)=\frac{2(n-α)}{n-2s}$ and $γ_{H}=4^s\frac{Γ^2(\frac{n+2s}{4})} {Γ^2(\frac{n-2s}{4})}$. We show that problem (\ref{eq0.1}) admits at least a weak solution under some conditions. To prove the main result, we develop some useful tools based on a weighted Morrey space. To be precise, we discover the embeddings \begin{equation} \label{eq0.2} {\dot{H}}^s(\R^{n}) \hookrightarrow {L}^{2^*_{s}(α)}(\R^{n},|y|^{-α}) \hookrightarrow L^{p,\frac{n-2s}{2}p+pr}(\R^{n},|y|^{-pr}) \end{equation} where $s \in (0,1)$, $0<α<2s<n$, $p\in[1,2^*_{s}(α))$, $r=\fracα{ 2^*_{s}(α) }$; We also establish an improved Sobolev inequality. By using mountain pass lemma along with an improved Sobolev inequality, we obtain a nontrivial weak solution to problem (\ref{eq0.1}) in a direct way. It is worth while to point out that the improved Sobolev inequality could be applied to simplify the proof of the main results in \cite{NGSS} and \cite{RFPP}.

math.AP↗

arXiv2018

Nondegeneracy of positive solutions to a Kirchhoff problem with critical Sobolev growth

In this paper, we prove uniqueness and nondegeneracy of positive solutions to the following Kirchhoff equations with critical growth \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)Δu=u^{5}, & u>0 & \text{in }\mathbb{R}^{3},\end{eqnarray*} where $a,b>0$ are positive constants. This result has potential applications in singular perturbation problems concerning Kirchhoff equaitons.

math.AP↗

arXiv2016

Uniqueness and nondegeneracy of positive solutions to a class of Kirchhoff equations in $\mathbb{R}^3$

In this paper, we establish a type of uniqueness and nondegeneracy results for positive solutions to the following nonlocal Kirchhoff equations \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\text{d} x\right)Δu+u=|u|^{p-1}u & & \text{in }\mathbb{R}^{3}, \end{eqnarray*} where $a,b$ are positive constants and $1<p<5$. Before this paper, it seems that there have no this type of results even on positive ground states solutions to Kirchhoff type equations, much less on general positive solutions. To overcome the difficulty brought by the nonlocality, some new observation on Kirchhoff equations is found, and some related theories on classical Schrödinger equations are applied.

math.AP↗

arXiv2014

Standing waves for a class of Schrödinger-Poisson equations in ${\mathbb{R}^3}$ involving critical Sobolev exponents

We are concerned with the following Schrödinger-Poisson equation with critical nonlinearity: \[\left\{\begin{gathered} - {\varepsilon ^2}Δu + V(x)u + ψu = λ|u{|^{p - 2}}u + |u{|^4}u{\text{in}}{\mathbb{R}^3}, \hfill - {\varepsilon ^2}Δψ= {u^2}{\text{in}}{\mathbb{R}^3},{\text{}}u > 0,{\text{}}u \in {H^1}({\mathbb{R}^3}), \hfill \end{gathered} \right. \] where $\varepsilon > 0$ is a small positive parameter, $λ> 0$, $3 < p \le 4$. Under certain assumptions on the potential $V$, we construct a family of positive solutions ${u_\varepsilon} \in {H^1}({\mathbb{R}^3})$ which concentrates around a local minimum of $V$ as $\varepsilon \to 0$.

math.AP↗