arXiv subjects

Xinfu Li

Publications and source records attributed to Xinfu Li.

15 recordsLinked to original sources

arXiv2026

Existence and multiplicity of solutions to the mean-field games model with mixed interactions

In this paper, we consider the stationary version of the Mean-Field Games (MFG) models. Inspired by \cite{Albuquerque-Silva2020, Bieganowski-Mederski2021, Lin-Wei05, Mederski-Schino2021}, we develop the minimization method on the Pohozaev manifold introduced in \cite{Soave20JDE, Soave20JFA} for the existence theory of the stationary version of the Mean-Field Games (MFG) models with $2$-homogeneous hamiltonians and mixed interactions. As applications, we prove the existence and multiplicity of radial solutions of the Mean-Field Games (MFG) models with general $p$-homogeneous hamiltonians and mixed interactions under more general conditions, some of which are even new for $2$-homogeneous hamiltonians. We hope that our techniques and ideas introduced in this paper would be helpful in understanding the optimal value of the total mass in the existence theory of radial solutions to the Mean-Field Games (MFG) models with general $p$-homogeneous hamiltonians and mixed interactions, as well as that of other models.

math.AP↗

arXiv2025

Asymptotic behavior of the least energy solutions to the Choquard equation in dimension two

In this paper, we are interested in the following planar Choquard equation \begin{equation*} \begin{cases} -\Delta u=\displaystyle\left(\int\limits_{\Omega}\frac{u^{p+1}(y)}{|x-y|^\alpha}dy\right)u^{p},\quad u>0,\ \ &\mbox{in}\ \Omega, \quad \ \ u=0, \ \ &\mbox{on}\ \partial \Omega, \end{cases} \end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^2$, $\alpha\in (0,2)$ and $p>1$ is a positive parameter. Unlike the higher-dimensional case, we prove that the least energy solutions $u_{p}$ neither blow up nor vanish, and develop only one peak as $p\to+\infty$ under suitable assumptions on $\Omega$. In contrast, the modified solutions $pu_p$ exhibit blow-up behavior analogous to that observed in higher dimensions. Furthermore, as $\alpha \to 0$, the main results of this paper become consistent with the known conclusions for the corresponding Lane-Emden equation.

math.AP↗

arXiv2025

Nondegeneracy of bubble solutions to the Choquard equation in two dimension

In this paper, we study the following Choquard equation with exponential nonlinearity \begin{equation*} -\Delta u=\left(\int_{\R^{2}}\frac{e^{u(y)}}{|x-y|^{\alpha}}dy\right)e^{u(x)},\quad \text{~in~}\R^{2}, \end{equation*} where $\alpha\in (0,2)$. Although the classification of solutions to this equation has been established recently, the nondegeneracy of its solutions remains open. Here, we prove the nondegeneracy by combining the integral representation of solutions with the spherical harmonic decomposition. The main result of this paper can be viewed as an extension of the nondegeneracy of solutions for both the planar Liouville equation and the higher-dimensional upper critical Choquard equation.

math.AP↗

arXiv2024

BeniFul: Backdoor Defense via Middle Feature Analysis for Deep Neural Networks

Backdoor defenses have recently become important in resisting backdoor attacks in deep neural networks (DNNs), where attackers implant backdoors into the DNN model by injecting backdoor samples into the training dataset. Although there are many defense methods to achieve backdoor detection for DNN inputs and backdoor elimination for DNN models, they still have not presented a clear explanation of the relationship between these two missions. In this paper, we use the features from the middle layer of the DNN model to analyze the difference between backdoor and benign samples and propose Backdoor Consistency, which indicates that at least one backdoor exists in the DNN model if the backdoor trigger is detected exactly on input. By analyzing the middle features, we design an effective and comprehensive backdoor defense method named BeniFul, which consists of two parts: a gray-box backdoor input detection and a white-box backdoor elimination. Specifically, we use the reconstruction distance from the Variational Auto-Encoder and model inference results to implement backdoor input detection and a feature distance loss to achieve backdoor elimination. Experimental results on CIFAR-10 and Tiny ImageNet against five state-of-the-art attacks demonstrate that our BeniFul exhibits a great defense capability in backdoor input detection and backdoor elimination.

cs.CR↗

arXiv2024

Existence of normalized solutions to Choquard equation with general mixed nonlinearities

We study the existence of normalized solutions to the following Choquard equation with $F$ being a Berestycki-Lions type function \begin{equation*} \begin{cases} -\Delta u+\lambda u=(I_{\alpha}\ast F(u))f(u),\quad \text{in}\ \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx=\rho^2, \end{cases} \end{equation*} where $N\geq 3$, $\rho>0$ is assigned, $\alpha\in (0,N)$, $I_{\alpha}$ is the Riesz potential, and $\lambda\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier. Here, the general nonlinearity $F$ contains the $L^2$-subcritical and $L^2$-supercritical mixed case, the Hardy-Littlewood-Sobolev lower critical and upper critical cases.

math.AP↗

arXiv2024

Existence of positive solutions for Kirchhoff type problems with critical exponent in exterior domains

In this paper, by using variational methods we study the existence of positive solutions for the following Kirchhoff type problem: $$ \left\{ \begin{array}{ll} -\left(a+b\mathlarger{\int}_{\Omega}|\nabla u|^{2}dx\right)\Delta u+V(x)u=u^{5}, \ & x\in\Omega,\\ \\ u=0,\ & x\in\partial \Omega, \end{array}\right. $$ where $a>0$, $b\geq0$, $\Omega\subset\mathbb R^3$ is an unbounded exterior domain, $\partial\Omega\neq\emptyset$, $\mathbb{R}^{3}\backslash\Omega$ is bounded, $u\in D_{0}^{1,2}(\Omega)$, and $V\in L^{\frac{3}{2}}(\Omega)$ is a non-negative continuous function. It turns out that the above Kirchhoff equation has no ground state solution. Nonetheless, by establishing some global compact lemma and constructing a suitable minimax value $c$ at a higher energy level where so called Palais-Smale condition holds, we succeed to obtain a positive solution for such a problem whenever $V$ and the hole $\mathbb{R}^{3}\setminus\Omega$ are suitable small in some senses. To the best of our knowledge, there are few similar results published in the literature concerning the existence of positive solutions for Kirchhoff equation in exterior domains. Our result also holds true in the case $\Omega=\mathbb R^3$, particularly, if $a=1$ and $b=0$, we improve some existing results (such as Benci, Cerami, Existence of positive solutions of the equation $-\Delta u+a(x)u=u^{(N+2)/(N-2)}$ in $\emph{R}^{N}$, J. Funct. Anal., 88 (1990), 90--117) for the corresponding Schr\"odinger equation in the whole space.

math.AP↗

arXiv2022

Normalized solutions to lower critical Choquard equation with a local perturbation

In this paper, we study the existence and non-existence of normalized solutions to the lower critical Choquard equation with a local perturbation \begin{equation*} \begin{cases} -Δu+λu=γ(I_α\ast|u|^{\frac{N+α}{N}})|u|^{\frac{N+α}{N}-2}u+μ|u|^{q-2}u,\quad \text{in}\ \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx=c^2, \end{cases} \end{equation*} where $γ, μ, c>0$, $2 0$, $2<q<2+\frac{4}{N}$, and $h$ is a positive and continuous function. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of $h$ when $ε$ is small enough.

math.AP↗

arXiv2022

Multiplicity and orbital stability of normalized solutions to non-autonomous Schrödinger equation with mixed nonlinearities

This paper studies the multiplicity of normalized solutions to the Schrödinger equation with mixed nonlinearities \begin{equation*} \begin{cases} -Δu=λu+h(εx)|u|^{q-2}u+η|u|^{p-2}u,\quad x\in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx=a^2, \end{cases} \end{equation*} where $a, ε, η>0$, $q$ is $L^2$-subcritical, $p$ is $L^2$-supercritical, $λ\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $h$ is a positive and continuous function. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of $h$ when $ε$ is small enough. Moreover, the orbital stability of the solutions obtained is analyzed as well. In particular, our results cover the Sobolev critical case $p=2N/(N-2)$.

math.AP↗

arXiv2021

Nonexistence, existence and symmetry of normalized ground states to Choquard equations with a local perturbation

We study the Choquard equation with a local perturbation \begin{equation*} -Δu=λu+(I_α\ast|u|^p)|u|^{p-2}u+μ|u|^{q-2}u,\ x\in \mathbb{R}^{N} \end{equation*} having prescribed mass \begin{equation*} \int_{\mathbb{R}^N}|u|^2dx=a^2. \end{equation*} For a $L^2$-critical or $L^2$-supercritical perturbation $μ|u|^{q-2}u$, we prove nonexistence, existence and symmetry of normalized ground states, by using the mountain pass lemma, the Pohožaev constraint method, the Schwartz symmetrization rearrangements and some theories of polarizations. In particular, our results cover the Hardy-Littlewood-Sobolev upper critical exponent case $p=(N+α)/(N-2)$. Our results are a nonlocal counterpart of the results in \cite{{Li 2021-4},{Soave JFA},{Wei-Wu 2021}}.

math.AP↗

arXiv2021

Studies of normalized solutions to Schrödinger equations with Sobolev critical exponent and combined nonlinearities

We consider the Sobolev critical Schrödinger equation with combined nonlinearities \begin{equation*} \begin{cases} -Δu=λu+|u|^{2^*-2}u+μ|u|^{q-2}u,\ \ x\in\mathbb{R}^{N},\\ u\in H^1(\mathbb{R}^N),\ \int_{\mathbb{R}^N}|u|^2dx=a, \end{cases} \end{equation*} where $N\geq 3$, $μ>0$, $λ\in \mathbb{R}$, $a>0$ and $q\in (2,2^*)$. We prove in this paper (1) Multiplicity and stability of solutions for $q\in (2,2+\frac{4}{N})$ and $μa^{\frac{q(1-γ_q)}{2}}\leq (2K)^{\frac{qγ_q-2^*}{2^*-2}}$ with $γ_q:=\frac{N}{2}-\frac{N}{q}$ and $K$ being some positive constant. This result extends the results obtained in Jeanjean et al. \cite{JEANJEAN-JENDREJ} and Jeanjean and Le \cite{Jeanjean-Le} for the case $μa^{\frac{q(1-γ_q)}{2}}<(2K)^{\frac{qγ_q-2^*}{2^*-2}}$ to the case $μa^{\frac{q(1-γ_q)}{2}}\leq (2K)^{\frac{qγ_q-2^*}{2^*-2}}$. (2) Nonexistence of ground states for $q=2+\frac{4}{N}$ and $μa^{\frac{q(1-γ_q)}{2}}\geq\bar{a}_N$ with $\bar{a}_N$ being some positive constant. We give a new proof to this result different with Wei and Wu \cite{Wei-Wu 2021}.

math.AP↗

arXiv2021

Standing waves to upper critical Choquard equation with a local perturbation: multiplicity, qualitative properties and stability

In this paper, we consider the upper critical Choquard equation with a local perturbation \begin{equation*} \begin{cases} -Δu=λu+(I_α\ast|u|^{p})|u|^{p-2}u+μ|u|^{q-2}u,\ x\in \mathbb{R}^{N},\\ u\in H^1(\mathbb{R}^N),\ \int_{\mathbb{R}^N}|u|^2=a, \end{cases} \end{equation*} where $N\geq 3$, $μ>0$, $a>0$, $λ\in \mathbb{R}$, $α\in (0,N)$, $p=\bar{p}:=\frac{N+α}{N-2}$, $q\in (2,2+\frac{4}{N})$ and $I_α=\frac{C}{|x|^{N-α}}$ with $C>0$. When $μa^{\frac{q(1-γ_q)}{2}}\leq (2K)^{\frac{qγ_q-2\bar{p}}{2(\bar{p}-1)}}$ with $γ_q=\frac{N}{2}-\frac{N}{q}$ and $K$ being some positive constant, we prove (1) Existence and orbital stability of the ground states. (2) Existence, positivity, radial symmetry, exponential decay and orbital instability of the ``second class' solutions. This paper generalized and improved parts of the results obtained in \cite{{JEANJEAN-JENDREJ},{Jeanjean-Le},{Soave JFA},{Wei-Wu 2021}} to the Schrödinger equation.

math.AP↗

arXiv2019

Orbital stability of standing waves for Schrödinger type equations with slowly decaying linear potential

In this paper, kinds of Schrödinger type equations with slowly decaying linear potential and power type or convolution type nonlinearities are considered. By using the concentration compactness principle, the sharp Gagliardo-Nirenberg inequality and a refined estimate of the linear operator, the existence and orbital stability of standing waves in $L^2$-subcritical and $L^2$-critical cases are established in a systematic way.

math.AP↗

arXiv2019

Global existence and blowup for Choquard equations with an inverse-square potential

In this paper, the Choquard equation with an inverse-square potential and both focusing and defocusing nonlinearities in the energy-subcritical regime is investigated. For all the cases, the local well-posedness result in $H^1(\mathbb{R}^N)$ is established. Moreover, the global existence result for arbitrary initial values is proved in the defocusing case while a global existence/blowup dichotomy below the ground state is established in the focusing case.

math.AP↗

arXiv2018

Groundstates for Choquard equations with the upper critical exponent

In this paper, an autonomous Choquard equation with the upper critical exponent is considered. By using the Pohožaev constraint method, the subcritical approximation method and the compactness lemma of Strauss, a groundstate solution in $H^1(\mathbb{R}^N)$ which is positive and radially symmetric is obtained. The result here extends and complements the earlier theorems.

math.AP↗

arXiv2018

Choquard equations with critical nonlinearities

In this paper, we study the Brezis-Nirenberg type problem for Choquard equations in $\mathbb{R}^N$ \begin{equation*} -Δu+u=(I_α\ast|u|^{p})|u|^{p-2}u+λ|u|^{q-2}u \quad \mathrm{in}\ \mathbb{R}^N, \end{equation*} where $N\geq 3,\ α\in(0,N)$, $λ>0$, $q\in (2,\frac{2N}{N-2}]$, $p=\frac{N+α}{N}$ or $\frac{N+α}{N-2}$ are the critical exponents in the sense of Hardy-Littlewood-Sobolev inequality and $I_α$ is the Riesz potential. Based on the results of the subcritical problems, and by using the subcritical approximation and the Pohožaev constraint method, we obtain a positive and radially nonincreasing groundstate solution in $H^1(\mathbb{R}^N)$ for the problem. To the end, the regularity and the Pohožaev identity of solutions to a general Choquard equation are obtained.

math.AP↗