arXiv subjects

Jinge Yang

Publications and source records attributed to Jinge Yang.

6 recordsLinked to original sources

arXiv2026

Existence and concentration of ground states to fractional Choquard equations

In this paper, we study the nonlinear fractional Choquard equation \begin{equation}\label{eq:0.1a} (-\Delta)^su+Vu=(|x|^{-\gamma}*|u|^2)u \quad {\rm in} \quad \mathbb{R}^N, \end{equation} where $0<\gamma<4$, $0<s<1$, $N\geq 4$ and $V\in C^1(\mathbb{R}^N)$ is a positive potential. Set $s_0=\frac {\gamma}{4}$. Under suitable assumptions on $V$, we prove that the equation admits a nonnegative ground state solution for $s\in(s_0,1)$, whereas no ground state solution exists for $0<s\le s_0$. Furthermore, we show that any ground state solution $u_s$ blows up and concentrates at a minimum point of $V$ as $s\downarrow s_0$. Finally, up to a subsequence, the ground state solution $u_s$ converges to a ground state solution of the classical Choquard equation as $s\uparrow 1$.

math.AP↗

arXiv2026

Asymptotic behavior of ground state solutions to nonlinear elliptic problems with the fractional Laplacian

In this paper, we consider the asymptotic behavior of the ground state solution $u_s$ of the nonlinear fractional Laplacian equation \begin{equation}\label{eq:0.1a} (-\Delta)^su+Vu=|u|^{p-2}u\quad x\in \mathbb{R}^n \end{equation} by taking $s$ as a parameter, where $n\geq 4$, $2<p<\frac{2n}{n-2}$, $V$ is a potential function. We show that for a fixed $p$, there exists $s_0\in(0,1)$ such that equation \eqref{eq:0.1a} admits a ground state solution $u_s$ if and only if $s_0<s<1$. Our main results give a description of the asymptotic behavior of $u_s$ as $s\uparrow1$ and $s\downarrow s_0$: $u_s$ converges to a function as $s\uparrow1$, and it blows up as $s\downarrow s_0$. Particularly, we prove that $u_s$ concentrates at a minimum point of the function $V$ as $s\downarrow s_0$. The local uniqueness of $u_s$ is also given.

math.AP↗

arXiv2024

Fractional Gross-Pitaevskii equations in non-Gaussian attractive Bose-Einstein condensates

In this paper, we investigate normalized solutions of a fractional Gross-Pitaevskii equation, which arises in an attractive Bose-Einstein condensation consisting of $N$ bosons moving by L\'{e}vy flights. We prove that there exists a positive constant $N^*$, such that if $0 N^*$ and $\alpha$ closed to $2$. We also study the asymptotic behavior of $u_\alpha$ and $v_\alpha$ as $\alpha\to 2_-$.

math.AP↗

arXiv2019

Normalized solutions and mass concentration for supercritical nonlinear Schr\"{o}dinger equations

In this paper, we deal with the existence and concentration of normalized solutions to the supercritical nonlinear Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{l} -\Delta u + V(x) u = \mu_q u + a|u|^q u \quad {\rm in}\quad \mathbb{R}^2,\\ \int_{\mathbb{R}^2}|u|^2\,dx =1,\\ \end{array} \right. \end{equation*} where $\mu_q$ is the Lagrange multiplier. We show that for $q>2$ close to $2$, the equation admits two solutions: one is the local minimal solution $u_q$ and another one is the mountain pass solution $v_q$. Furthermore, we study the limiting behavior of $u_q$ and $v_q$ when $q\to 2_+$. Particularly, we describe precisely the blow-up formation of the excited state $v_q$.

math.AP↗

arXiv2019

On supercritical nonlinear Schr\"{o}dinger equations with ellipse-shaped potentials

In this paper, we study the existence and concentration of normalized solutions to the supercritical nonlinear Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{l} -\Delta u + V(x) u = \mu_q u + a|u|^q u \quad {\rm in}\quad \mathbb{R}^2,\\ \int_{\mathbb{R}^2}|u|^2\,dx =1,\\ \end{array} \right. \end{equation*} where $\mu_q$ is the Lagrange multiplier. For ellipse-shaped potentials $V(x)$, we show that for $q>2$ close to $2$, the equation admits an excited solution $u_q$, and furthermore, we study the limiting behavior of $u_q$ when $q\to 2_+$. Particularly, we describe precisely the blow-up formation of the excited state $u_q$.

math.AP↗

arXiv2017

Existence and mass concentration of pseudo-relativistic Hartree equation

In this paper, we investigate the constrained minimization problem \begin{equation}\label{eq:0.1} e(a):=\inf_{\{u\in \mathcal{H},\|u\|_2^2=1\}}E_a(u), \end{equation} where the energy functional \begin{equation} \label{eq:0.2} E_a(u)=\int_{\mathbb{R}^3}(u\sqrt{-\Delta+m^2}\,u+Vu^2)\,dx -\frac{a}{2}\int_{\mathbb{R}^3}(|x|^{-1}*u^2)u^2\,dx \end{equation} with $m\in \mathbb{R}$, $a>0$, is defined on a Sobolev space $\mathcal{H}$. We show that there exists a threshold $a^*>0$ so that $e(a)$ is achieved if $0<a<a^*$, and has no minimizers if $a\geq a^*$. We also investigate the asymptotic behavior of nonnegative minimizers of $e(a)$ as $a\to a^*$.

math.AP↗