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Jianfu Yang

Publications and source records attributed to Jianfu Yang.

17 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↗
arXiv2023

Infinitely many new solutions for singularly perturbed Schrödinger equations

This paper deals with the existence of solutions for the following perturbed Schrödinger equation \begin{equation*} -\varepsilon^{2} Δu + V(x)u= |u|^{p-2}u, \, \, \text{ in } \, \, \r^{N}, \end{equation*} where $\varepsilon$ is a parameter, $N \geq 3$, $p \in (2, \frac{2N}{N-2})$, and $V(x)$ is a potential function in $\r^{N}$. We demonstrate an interesting ``dichotomy'' phenomenon for concentrating solutions of the above Schrödinger equation. More specifically, we construct infinitely many new solutions with peaks locating both in the bounded domain and near infinity, which fulfills the profile of the concentration compactness. Moreover, this approach can be extended to solve other related problems.

math.AP↗
arXiv2022

Positive solutions to multi-critical Schrödinger equations

In this paper, we investigate the existence of multiple positive solutions to the following multi-critical Schrödinger equation \begin{equation} \label{p} \begin{cases} -Δu+λV(x)u=μ|u|^{p-2}u+\sum\limits_{i=1}^{k}(|x|^{-(N-α_i)}* |u|^{2^*_i})|u|^{2^*_i-2}u\quad \text{in}\ \mathbb{R}^N,\\ \qquad\qquad\qquad u\,\in H^1(\mathbb{R}^N), \end{cases} \end{equation} where $λ,μ\in \mathbb{R}^+, \, N\geqslant 4$, and $2^*_i=\frac{N+α_i}{N-2}$ with $N-4<α_i<N,\,i=1,2,\cdots,k$ are critical exponents and $2<p<2^*_{min}=\min\{2^*_i:i=1,2,\cdots,k\}$. Suppose that $Ω=int\,V^{-1}(0)\subset\mathbb{R}^N$ is a bounded domain, we show that for $λ$ large, problem above possesses at least $cat_Ω(Ω)$ positive solutions.

math.AP↗
arXiv2022

Positive solutions to multi-critical elliptic problems

In this paper, we investigate the existence of multiple solutions to the following multi-critical elliptic problem \begin{equation}\label{eq:0.1} \left\{\begin{aligned} -Δu & =λ|u|^{p-2}u +\sum_{i=1}^k(|x|^{-(N-α_i)}*|u|^{2^*_i})|u|^{2^*_i-2}u\quad {\rm in}\quad Ω,\\ &u\in H^1_0(Ω)\\ \end{aligned}\right. \end{equation} in connection with the topology of the bounded domain $Ω\subset \mathbb{R}^N, \,N\geq 4$, where $λ>0$, $2^*_i=\frac{N+α_i}{N-2}$ with $N-4<α_i 0$ such that if $0<λ<λ^*$ problem \eqref{eq:0.1} possesses at least $cat_Ω(Ω)$ positive solutions. We also study the existence and uniqueness of solutions for the limit problem of \eqref{eq:0.1}.

math.AP↗
arXiv2020

Multi-Peak solutions to Chern-Simons-Schrödinger systems with non-radial potential

In this paper, we consider the existence of static solutions to the nonlinear Chern-Simons-Schrödinger system \begin{equation}\label{eqabstr} \left\{\begin{array}{ll} -ihD_0Ψ-h^2(D_1D_1+D_2D_2)Ψ+VΨ=|Ψ|^{p-2}Ψ,\\ \partial_0A_1-\partial_1A_0=-\frac 12ih[\overlineΨD_2Ψ-Ψ\overline{D_2Ψ}],\\ \partial_0A_2-\partial_2A_0=\frac 12ih[\overlineΨD_1Ψ-Ψ\overline{D_1Ψ}],\\ \partial_1A_2-\partial_2A_1=-\frac12|Ψ|^2,\\ \end{array} \right. \end{equation} where $p>2$ and non-radial potential $V(x)$ satisfies some certain conditions. We show that for every positive integer $k$, there exists $h_0>0$ such that for $0<h<h_0$, problem \eqref{eqabstr} has a nontrivial static solution $(Ψ_h, A_0^h, A_1^h,A_2^h)$. Moreover, $Ψ_h$ is a positive non-radial function with $k$ positive peaks, which approach to the local maximum point of $V(x)$ as $h\to 0^+$.

math.AP↗
arXiv2020

Quasi-Direct Drive Actuation for a Lightweight Hip Exoskeleton with High Backdrivability and High Bandwidth

High-performance actuators are crucial to enable mechanical versatility of lower-limb wearable robots, which are required to be lightweight, highly backdrivable, and with high bandwidth. State-of-the-art actuators, e.g., series elastic actuators (SEAs), have to compromise bandwidth to improve compliance (i.e., backdrivability). In this paper, we describe the design and human-robot interaction modeling of a portable hip exoskeleton based on our custom quasi-direct drive (QDD) actuation (i.e., a high torque density motor with low ratio gear). We also present a model-based performance benchmark comparison of representative actuators in terms of torque capability, control bandwidth, backdrivability, and force tracking accuracy. This paper aims to corroborate the underlying philosophy of "design for control", namely meticulous robot design can simplify control algorithms while ensuring high performance. Following this idea, we create a lightweight bilateral hip exoskeleton (overall mass is 3.4 kg) to reduce joint loadings during normal activities, including walking and squatting. Experimental results indicate that the exoskeleton is able to produce high nominal torque (17.5 Nm), high backdrivability (0.4 Nm backdrive torque), high bandwidth (62.4 Hz), and high control accuracy (1.09 Nm root mean square tracking error, i.e., 5.4% of the desired peak torque). Its controller is versatile to assist walking at different speeds (0.8-1.4 m/s) and squatting at 2 s cadence. This work demonstrates significant improvement in backdrivability and control bandwidth compared with state-of-the-art exoskeletons powered by the conventional actuation or SEA.

cs.RO↗
arXiv2019

Normalized solutions and mass concentration for supercritical nonlinear Schrödinger equations

In this paper, we deal with the existence and concentration of normalized solutions to the supercritical nonlinear Schrödinger equation \begin{equation*} \left\{ \begin{array}{l} -Δu + V(x) u = μ_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 $μ_q$ 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 Schrödinger equations with ellipse-shaped potentials

In this paper, we study the existence and concentration of normalized solutions to the supercritical nonlinear Schrödinger equation \begin{equation*} \left\{ \begin{array}{l} -Δu + V(x) u = μ_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 $μ_q$ 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

Multiple nodal solutions of nonlinear Choquard equations

In this paper, we consider the existence of multiple nodal solutions of the nonlinear Choquard equation \begin{equation*} \ \ \ \ (P)\ \ \ \ \begin{cases} -Δu+u=(|x|^{-1}\ast|u|^p)|u|^{p-2}u \ \ \ \text{in}\ \mathbb{R}^3, \ \ \ \ \\ u\in H^1(\mathbb{R}^3),\\ \end{cases} \end{equation*} where $p\in (\frac{5}{2},5)$. We show that for any positive integer $k$, problem $(P)$ has at least a radially symmetrical solution changing sign exactly $k$-times.

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{-Δ+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↗
arXiv2015

Weak solutions of semilinear elliptic equation involving Dirac mass

In this paper, we study the following elliptic problem with Dirac mass \begin{equation}\label{eq 0.1} -Δu=Vu^p+k δ_0\quad {\rm in}\quad \mathbb{R}^N, \qquad \lim_{|x|\to+\infty}u(x)=0, \end{equation} where $N>2$, $p>0$, $k>0$, $δ_0$ is Dirac mass at the origin, the function $V$ is a locally Lipchitz continuous in $\mathbb{R}^N\setminus\{0\}$ satisfying $$ V(x)\le \frac{c_1}{|x|^{a_0}(1+|x|^{a_\infty-a_0})} $$ with $a_0 a_0 $ and $c_1>0$. We obtain two positive solutions of (\ref{eq 0.1}) with additional conditions for parameters on $a_\infty, a_0$, $p$ and $k$. The first solution is a minimal positive solution and the second solution is constructed by Mountain Pass theorem.

math.AP↗
arXiv2015

Equations involving fractional Laplacian operator: Compactness and application

In this paper, we consider the following problem involving fractional Laplacian operator: \begin{equation}\label{eq:0.1} (-Δ)^α u= |u|^{2^*_α-2-\varepsilon}u + λu\,\, {\rm in}\,\, Ω,\quad u=0 \,\, {\rm on}\, \, \partialΩ, \end{equation} where $Ω$ is a smooth bounded domain in $\mathbb{R}^N$, $\varepsilon\in [0, 2^*_α-2)$, $0<α<1,\, 2^*_α= \frac {2N}{N-2α}$. We show that for any sequence of solutions $u_n$ of \eqref{eq:0.1} corresponding to $\varepsilon_n\in [0, 2^*_α-2)$, satisfying $\|u_n\|_{H}\le C$ in the Sobolev space $H$ defined in \eqref{eq:1.1a}, $u_n$ converges strongly in $H$ provided that $N>6α$ and $λ>0$. An application of this compactness result is that problem \eqref{eq:0.1} possesses infinitely many solutions under the same assumptions.

math.AP↗
arXiv2015

Fractional Hardy-Sobolev elliptic problems

In this paper, we study the following singular nonlinear elliptic problem \begin{equation}\label{eq:1} \left\{ \begin{array}{ll} \displaystyle (-Δ)^{\frac α2} u=λ|u|^{r-2}u+μ\frac{|u|^{q-2}u}{|x|^{s}}\quad &{\rm in }\quad Ω, \\ \\ u=0 &{\rm on }\quad \partialΩ, \end{array} \right. \end{equation} where $Ω$ is a smooth bounded domain in $\mathbb R^N$ with $0\in Ω$, $λ,μ>0,0<s\leqα$, $(-Δ)^{\frac α2}$ is the fractional Laplacian operator with $0<α<2$. We establish existence results of problem \eqref{eq:1} for subcritical, Sobolev critical and Hardy-Sobolev critical cases.

math.AP↗
arXiv2014

Semilinear fractional elliptic equations with measures in unbounded domain

In this paper, we study the existence of nonnegative weak solutions to (E) $ (-Δ)^αu+h(u)=ν$ in a general regular domain $Ω$, which vanish in $\R^N\setminusΩ$, where $(-Δ)^α$ denotes the fractional Laplacian with $α\in(0,1)$, $ν$ is a nonnegative Radon measure and $h:\mathbb{R}_+\to\mathbb{R}_+$ is a continuous nondecreasing function satisfying a subcritical integrability condition. Furthermore, we analyze properties of weak solution $u_k$ to $(E)$ with $Ω=\mathbb{R}^N$, $ν=kδ_0$ and $h(s)=s^p$, where $k>0$, $p\in(0,\frac{N}{N-2α})$ and $δ_0$ denotes Dirac mass at the origin. Finally, we show for $p\in(0,1+\frac{2α}{N}]$ that $u_k\to\infty$ in $\mathbb{R}^N$ as $k\to\infty$, and for $p\in(1+\frac{2α}{N},\frac{N}{N-2α})$ that $\lim_{k\to\infty}u_k(x)=c|x|^{-\frac{2α}{p-1}}$ with $c>0$, which is a classical solution of $ (-Δ)^αu+u^p=0$ in $\mathbb{R}^N\setminus\{0\}$.

math.AP↗