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- Multiplicity and Concentration of Positive Solutions for a Quasilinear Schrödinger-Poisson System with Critical Nonlinearity
- ZHANG Weiqiang, WEN Yanyun
- 2025, 45(1): 1-24. doi: 10.3969/j.issn.1006-8074.2025.01.001
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Abstract
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1058 )
PDF (258KB)
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623
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In this paper, we study the following Schr\"odinger-Poisson system
\begin{equation*}
\left\{
\begin{array}{ll}
-\varepsilon^{p}\Delta_{p}u+V(x)|u|^{p-2}u+\phi |u|^{p-2}u=f(u)+|u|^{p^{*}-2}u\quad \mbox{in}\quad \mathbb{R}^{3}, \\
-\varepsilon^{2}\Delta \phi =|u|^{p}\quad\mbox{in}\quad \mathbb{R}^{3},
\end{array}
\right.
\end{equation*}
where $\varepsilon>0$ is a parameter, $\frac{3}{2}<p<3$, $\Delta_{p}u=\text{div}(|\nabla u|^{p-2}\nabla u)$, $p^{*}=\frac{3p}{3-p}$, $V:\mathbb{R}^{3}\rightarrow\mathbb{R}$ is a potential function with a local minimum and $f$ is subcritical growth. Based on the penalization method, Nehari manifold techniques and Ljusternik-Schnirelmann category theory, we obtain the multiplicity and concentration of positive solutions to the above system.
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Sign-changing Solutions for a Fractional Schrödinger-Poisson System with Concave-convex Nonlinearities and a Steep Potential Well
- FU Jiao, LI Hongying, LIAO Jiafeng
- 2025, 45(1): 25-44. doi: 10.3969/j.issn.1006-8074.2025.01.002
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Abstract
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1019 )
PDF (240KB)
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172
)
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In this paper, we investigate the following fractional Schrödinger-Poisson system with concave-convex nonlinearities and steep potential well
\begin{equation}
\begin{cases}
(-\Delta)^s u+V_{\lambda} (x)u+\phi u=f(x)|u|^{q-2}u+|u|^{p-2}u, & ~\mathrm{in}~~\mathbb{R}^3, \\
(-\Delta)^t \phi=u^2, & ~\mathrm{in}~~\mathbb{R}^3,
\end{cases}
\nonumber
\end{equation}
where $s\in(\frac{3}{4},1), t\in(0,1)$, $q\in(1,2)$, $p\in(4,2_s^*)$, $2_s^*:=\frac{6}{3-2s}$ is the fractional critical exponent in dimension 3, $V_{\lambda}(x)$ = $\lambda V(x)+1$ with $\lambda>0$. Under the case of steep potential well, we obtain the existence of the sign-changing solutions for the above system by using the constraint variational method and the quantitative deformation lemma. Furthermore, we prove that the energy of ground state sign-changing solution is strictly more than twice of the energy of the ground state solution. Our results improve the recent results in the literature.
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Normalized Solutions for p-Laplacian Schrödinger-Poisson Equations with L2-supercritical Growth
- 2025, 45(1): 45-61. doi: 10.3969/j.issn.1006-8074.2025.01.003
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Abstract
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1051 )
PDF (222KB)
(
211
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In this paper, we consider the $p$-Laplacian Schrödinger-Poisson equation with $L^{2}$-norm constraint
$$-\Delta_p u+|u|^{p-2}u+\lambda u+ \left(\frac{1}{4\pi|x|}*|u|^2\right)u=|u|^{q-2} u,\and x \in \mathbb{R}^3,$$
where $2 \leq p<3$, $\frac{5 p}{3}<q<p^{*}=\frac{3p}{3-p}$, $\lambda>0$ is a Lagrange multiplier. We obtain the critical point of the corresponding functional of the problem on mass constraint by the variational method and the Mountain pass lemma, and then find a normalized solution to this equation.
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- Direct and Inverse Problems for a Third-order Differential Operator with Anti-periodic Boundary Conditions and a Non-local Potential#br#
- ZHANG Mingming, LIU Yixuan
- 2025, 45(1): 62-80. doi: 10.3969/j.issn.1006-8074.2025.01.004
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Abstract
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974 )
PDF (212KB)
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245
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- This paper focuses on the direct and inverse problems for a third-order self-adjoint differential operator with non-local potential and anti-periodic boundary conditions. Firstly, we obtain the expressions for the characteristic function and resolvent of this third-order differential operator. Secondly, by using the expression for the resolvent of the operator, we prove that the spectrum for this operator consists of simple eigenvalues and a finite number of eigenvalues with multiplicity 2. Finally, we solve the inverse problem for this operator, which states that the non-local potential function can be reconstructed from four spectra. Specially, we prove the Ambarzumyan theorem and indicate that odd or even potential functions can be reconstructed by three spectra.
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Positive Periodic Solutions to a Second-order Nonlinear Differential Equation with an Indefinite Singularity
- YUAN Shujing, LI Shaowen, CHENG Zhibo
- 2025, 45(1): 81-93. doi: 10.3969/j.issn.1006-8074.2025.01.005
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Abstract
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924 )
PDF (193KB)
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161
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In this paper, we provide new sufficient conditions for the existence of positive periodic solutions for a class of indefinite singular differential equation
\begin{equation*}
x''(t)+a(t)x(t)=\frac{h(t)}{x^\rho(t)}+g(t)x^\delta(t)+e(t),
\end{equation*}
where $\rho$ and $\delta$ are two positive constants and
$0<\delta\leq1$, $~h,~e\in
L^1(\mathbb{R}/T\mathbb{Z})$, $g\in L^1(\mathbb{R}/T\mathbb{Z})$ is
positive. Our proofs are based on the fixed point theorems (Schauder's fixed
point theorem and Krasnoselski$\breve{\mbox{i}}$-Guo's fixed point
theorem) and the positivity of the associated Green function.
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- Green's Functions of a Class of Sturm-Liouville Problems with Eigenparameters in Boundary Conditions
- XIAO Lu, MU Dan, ZHANG Ruifan
- 2025, 45(1): 94-106. doi: 10.3969/j.issn.1006-8074.2025.01.006
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Abstract
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963 )
PDF (165KB)
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248
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- In this paper, we study a class of Sturm-Liouville problem, where the boundary conditions involve eigenparameters. Firstly, by defining a new inner product which depends on the transmission conditions, we obtain a new Hilbert space, on which the concerned operator $A$ is self-adjoint. Then we construct the fundamental solutions to the problem, obtain the necessary and sufficient conditions for eigenvalues, and prove that the eigenvalues are simple. Finally, we investigated Green's functions of such problem.
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Some Topological Indices of Nil-clean Graphs of ℤn
- SU Huadong, LIANG Zhunti
- 2025, 45(1): 107-114. doi: 10.3969/j.issn.1006-8074.2025.01.007
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Abstract
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932 )
PDF (151KB)
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683
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- In this paper, we determine the Sombor index and some other vertex-degree-based topological indices for the nil-clean graphs of $\mathbb{Z}_n$, the ring of integers modulo $n$.
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Anticipating Lag Synchronization Based on Machine Learning
- WU Yongqing, BAO Xingxing
- 2025, 45(1): 115-126. doi: 10.3969/j.issn.1006-8074.2025.01.008
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Abstract
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1017 )
PDF (1529KB)
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404
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This paper propose a comprehensive data-driven prediction framework based on machine learning methods to investigate the lag synchronization phenomenon in coupled chaotic systems, particularly in cases where accurate mathematical models are challenging to establish or where system equations remain unknown. The Long Short-Term Memory (LSTM) neural network is trained using time series acquired from the desynchronization system states, subsequently predicting the lag synchronization transition. In the experiments, we focus on the Lorenz system with time-varying delayed coupling, studying the effects of coupling coefficients and time delays on lag synchronization, respectively. The results indicate that with appropriate training, the machine learning model can adeptly predict the lag synchronization occurrence and transition. This study not only enhances our comprehension of complex network synchronization behaviors but also underscores the potential and practical applications of machine learning in exploring nonlinear dynamic systems.
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