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Chao Ji

Publications and source records attributed to Chao Ji.

At least 37 records · Page 2Linked to original sources
arXiv2025

Normalized solutions of nonlinear Dirac equations on noncompact metric graphs with localized nonlinearities

In this paper, we study the following nonlinear Dirac equations (NLDE) on noncompact metric graph $\mathcal{G}$ with localized nonlinearities \begin{equation} \mathcal{D} u - ωu= aχ_{\mathcal{K}}|u|^{p-2}u, \end{equation} where $\mathcal{D}$ is the Dirac operator on $\mathcal{G}$, $u: \mathcal{G} \to \mathbb{C}^2$, $ω\in \mathbb{R}$, $a > 0$, $χ_{\mathcal{K}}$ is the characteristic function of the compact core $\mathcal{K}$, and $p>2$. First, for $2 2$, prove the existence of normalized solutions to (NLDE) when $λ= -mc^2$ is an eigenvalue of the operator $\mathcal{D}$. In the Appendix, we study the influence of the parameters $m, c > 0$ on the existence of normalized solutions to (NLDE). To the best of our knowledge, this is the first study to investigate the normalized solutions to (NLDE) on metric graphs.

math.AP↗
arXiv2025

Existence and multiplicity of normalized solutions for $(2,q)$-Laplacian equations with generic double-behaviour nonlinearities

In this paper, we study {existence and multiplicity} of normalized solutions for the following $(2, q)$-Laplacian equation \begin{equation*}\label{Eq-Equation1} \left\{\begin{array}{l} -Δu-Δ_q u+λu=f(u) \quad x \in \mathbb{R}^N , \int_{\mathbb{R}^N}u^2 d x=c^2, \end{array}\right. \end{equation*} where $1 0$ is a constant. The nonlinearity $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous, with mass-subcritical growth at the origin, mass-supercritical growth at infinity, and is more general than the sum of two powers. Under different assumptions, we prove the existence of a locally least-energy solution and the existence of a second solution with higher energy.

math.AP↗
arXiv2025

Normalized solutions to a class of $(2, q)$-Laplacian equationsin the strongly sublinear regime

In this paper, we consider the existence and multiplicity of normalized solutions for the following $(2, q)$-Laplacian equation \begin{equation}\label{Equation1} \left\{\begin{aligned} &-Δu-Δ_q u+λu=g(u),\quad x \in \mathbb{R}^N, &\int_{\mathbb{R}^N}u^2 d x=c^2, \end{aligned}\right. \tag{$\mathscr E_λ$} \end{equation} where $1 0$ is a constant. The nonlinearity $g:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and the behaviour of $g$ at the origin is allowed to be strongly sublinear, i.e., $\lim \limits _{s \rightarrow 0} g(s) / s=-\infty$, which includes the logarithmic nonlinearity $$ g(s)= s \log s^2. $$ We consider a family of approximating problems that can be set in $H^1\left(\mathbb{R}^N\right)\cap D^{1, q}\left(\mathbb{R}^N\right)$ and the corresponding least-energy solutions. Then, we prove that such a family of solutions converges to a least-energy solution to the original problem. Additionally, under certain assumptions about $g$ that allow us to work in a suitable subspace of $H^1\left(\mathbb{R}^N\right)\cap D^{1, q}\left(\mathbb{R}^N\right)$, we prove the existence of infinitely many solutions of the above $(2, q)$-Laplacian equation.

math.AP↗
arXiv2024

The double phase problems on lattice graphs

In this paper, we first develop the theory of Musielak-Orlicz spaces on locally finite graphs, including completeness, reflexivity, separability, and so on. Then, we give some elementary properties of double phase operators on locally finite graphs. Finally, as applications of previous theory, we prove some existence results of solutions to double phase problems on lattice graphs.

math.AP↗
arXiv2024

Deep Reinforcement Learning in Autonomous Car Path Planning and Control: A Survey

Combining data-driven applications with control systems plays a key role in recent Autonomous Car research. This thesis offers a structured review of the latest literature on Deep Reinforcement Learning (DRL) within the realm of autonomous vehicle Path Planning and Control. It collects a series of DRL methodologies and algorithms and their applications in the field, focusing notably on their roles in trajectory planning and dynamic control. In this review, we delve into the application outcomes of DRL technologies in this domain. By summarizing these literatures, we highlight potential challenges, aiming to offer insights that might aid researchers engaged in related fields.

cs.RO↗
arXiv2024

Existence and multiplicity of solutions for the logarithmic Schrödinger equation with a potential on lattice graphs

In this paper, we consider the existence and multiplicity of solutions for the logarithmic Schrödinger equation on lattice graphs $\mathbb{Z}^N$ $$ -Δu+V(x) u=u \log u^2, \quad x \in \mathbb{Z}^N, $$ When the potential $V$ is coercive, we obtain infinitely many solutions by adapting some arguments of the Fountain theorem. In the cases of periodic potential, asymptotically periodic potential and bounded potential, we first investigate the existence of ground state solutions via the variation methods, and then we generalize these results from $\mathbb{Z}^N$ to quasi-transitive graphs. Finally, we extend the main results of the paper to the $p$-Laplacian equation with the logarithmic nonlinearity.

math.AP↗
arXiv2023

Nodal solutions for the double phase problems

We consider a parametric nonautonomous $(p, q)$-equation with unbalanced growth as follows \begin{align*} \left\{ \begin{aligned} &-Δ_p^αu(z)-Δ_q u(z)=λ\vert u(z)\vert^{τ-2}u(z)+f(z, u(z)), \quad \quad \hbox{in }Ω,\\ &u|_{\partial Ω}=0, \end{aligned} \right. \end{align*} where $Ω\subseteq \mathbb{R}^N$ be a bounded domain with Lispchitz boundary $\partialΩ$, $α\in L^{\infty}(Ω)\backslash \{0\}$, $a(z)\geq 0$ for a.e. $z \in Ω$, $ 1<τ< q 0$. In the reaction there is a parametric concave term and a perturbation $f(z, x)$. Under the minimal conditions on $f(z, 0)$, which essentially restrict its growth near zero, by employing variational tools, truncation and comparison techniques, as well as critical groups, we prove that for all small values of the parameter $λ>0$, the problem has at least three nontrivial bounded solutions (positive, negative, nodal), which are ordered and asymptotically vanish as $λ\rightarrow 0^{+}$.

math.AP↗
arXiv2023

Multiple normalized solutions to a logarithmic Schrödinger equation via Lusternik-Schnirelmann category

In this paper our objective is to investigate the existence of multiple normalized solutions to the logarithmic Schrödinger equation given by \begin{align*} \left\{ \begin{aligned} &-ε^2 Δu+V( x)u=λu+u \log u^2, \quad \quad \hbox{in }\mathbb{R}^N,\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2}ε^N, \end{aligned} \right. \end{align*} where $a, ε>0, λ\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier and $V: \mathbb{R}^N \rightarrow[-1, \infty)$ is a continuous function. Our analysis demonstrates that the number of normalized solutions of the equation is associated with the topology of the set where the potential function $V$ attains its minimum value. To prove the main result, we employ minimization techniques and use the Lusternik-Schnirelmann category. Additionally, we introduce a new function space where the energy functional associated with the problem is of class $C^1$.

math.AP↗
arXiv2021

Semiclassical states for a magnetic nonlinear Schrödinger equation with exponential critical growth in $\mathbb{R}^{2}$

This paper is devoted to the magnetic nonlinear Schrödinger equation \[ \Big(\frac{\varepsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u=f(| u|^{2})u \text{ in } \mathbb{R}^{2}, \] where $\varepsilon>0$ is a parameter, $V:\mathbb{R}^{2}\rightarrow \mathbb{R}$ and $A: \mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ are continuous functions and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^{1}$ function having exponential critical growth. Under a global assumption on the potential $V$, we use variational methods and Ljusternick-Schnirelmann theory to prove existence, multiplicity, concentration, and decay of nontrivial solutions for $\varepsilon>0$ small.

math.AP↗
arXiv2021

Normalized solutions for a Schrödinger equation with critical growth in $\mathbb{R}^{N}$

In this paper we study the existence of normalized solutions to the following nonlinear Schrödinger equation with critical growth \begin{align*} \left\{ \begin{aligned} &-Δu=λu+f(u), \quad \quad \hbox{in }\mathbb{R}^N,\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2}, \end{aligned} \right. \end{align*} where $a>0$, $λ\in \mathbb{R}$ and $f$ has an exponential critical growth when $N=2$, and $f(t)=μ|u|^{q-2}u+|u|^{2^*-2}u$ with $q \in (2+\frac{4}{N},2^*)$, $μ>0$ and $2^*=\frac{2N}{N-2}$ when $N \geq 3$. Our main results complement some recent results for $N \geq 3$ and it is totally new for $N=2$.

math.AP↗
arXiv2021

Multiplicity of normalized solutions for a Schrödinger equation with critical growth in $\mathbb{R}^{N}$

In this paper we study the multiplicity of normalized solutions to the following nonlinear Schrödinger equation with critical growth \begin{align*} \left\{ \begin{aligned} &-Δu=λu+μ|u|^{q-2}u+f(u), \quad \quad \hbox{in }\mathbb{R}^N,\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2}, \end{aligned} \right. \end{align*} where $a,μ>0$, $λ\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $q \in (2,2+\frac{4}{N})$ and $f$ has an exponential critical growth when $N=2$, and $f(u)=|u|^{2^*-2}u$ when $N \geq 3$ and $2^{*}=\frac{2N}{N-2}$.

math.AP↗
arXiv2020

Multi-bump positive solutions for a logarithmic Schrödinger equation with deepening potential well

This article concerns the existence of multi-bump positive solutions for the following logarithmic Schrödinger equation $$ \left\{ \begin{array}{lc} -Δu+ λV(x)u=u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \\ u \in H^1(\mathbb{R}^{N}), \\ \end{array} \right. $$ where $N \geq 1$, $λ>0$ is a parameter and the nonnegative continuous function $V: \mathbb{R}^{N}\rightarrow \mathbb{R}$ has a potential well $Ω: =\text{int}\, V^{-1}(0)$ which possesses $k$ disjoint bounded components $Ω=\bigcup_{j=1}^{k}Ω_{j}$. Using the variational methods, we prove that if the parameter $λ>0$ is large enough, then the equation has at least $2^{k}-1$ multi-bump positive solutions.

math.AP↗
arXiv2020

Multiplicity and concentration results for a magnetic Schrödinger equation with exponential critical growth in $\mathbb{R}^{2}$

In this paper we study the following nonlinear Schrödinger equation with magnetic field \[ \Big(\frac{\varepsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u=f(| u|^{2})u,\quad x\in\mathbb{R}^{2}, \] where $\varepsilon>0$ is a parameter, $V:\mathbb{R}^{2}\rightarrow \mathbb{R}$ and $A: \mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ are continuous potentials and $f:\mathbb{R}\rightarrow \mathbb{R}$ has exponential critical growth. Under a local assumption on the potential $V$, by variational methods, penalization technique, and Ljusternick-Schnirelmann theory, we prove multiplicity and concentration of solutions for $\varepsilon$ small.

math.AP↗
arXiv2020

Existence and concentration of positive solutions for a logarithmic Schrödinger equation via penalization method

In this article we are concerned with the following logarithmic Schrödinger equation $$ \left\{ \begin{array}{lc} -ε^2Δu+ V(x)u=u \log u^2, & \mbox{in} \,\, \mathbb{R}^{N}, \\ %u(x)>0, & \mbox{in} \quad \mathbb{R}^{N} \\ u \in H^1(\mathbb{R}^{N}), & \; \\ \end{array} \right. $$ where $ε>0, N \geq 1$ and $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential. Under a local assumption on the potential $V$, we use the variational methods to prove the existence and concentration of positive solutions for the above problem.

math.AP↗
arXiv2019

Multiple positive solutions for a Schrödinger logarithmic equation

This article concerns with the existence of multiple positive solutions for the following logarithmic Schrödinger equation $$ \left\{ \begin{array}{lc} -ε^2Δu+ V(x)u=u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \\ %u(x)>0, & \mbox{in} \quad \mathbb{R}^{N} \\ u \in H^1(\mathbb{R}^{N}), & \; \\ \end{array} \right. $$ where $ε>0$, $N \geq 1$ and $V$ is a continuous function with a global minimum. Using variational method, we prove that for small enough $ε>0$, the "shape" of the graph of the function $V$ affects the number of nontrivial solutions.

math.AP↗
arXiv2019

Existence of a positive solution for a logarithmic Schrödinger equation with saddle-like potential

In this article we use the variational method developed by Szulkin \cite{szulkin} to prove the existence of a positive solution for the following logarithmic Schrödinger equation $$ \left\{ \begin{array}{lc} -ε^2Δu+ V(x)u=u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \\ %u(x)>0, & \mbox{in} \quad \mathbb{R}^{N} \\ u \in H^1(\mathbb{R}^{N}), & \; \\ \end{array} \right. $$ where $ε>0, N \geq 1$ and $V$ is a saddle-like potential.

math.AP↗
arXiv2018

The cone Moser-Trudinger inequalities and their applications

In this article, we firstly study the cone Moser-Trudinger inequalities and their best exponents $α_2$ on both bounded and unbounded domains $\mathbb{R}^2_{+}$. Then, using the cone Moser-Trudinger inequalities, we study the existence of weak solutions to the nonlinear equation \begin{equation*} \left\{\begin{array}{ll} -Δ_{\mathbb{B}} u=f(x, u), &\mbox{in}\ x\in \mbox{int} (\mathbb{B}), \\ u= 0, &\mbox{on}\ \partial\mathbb{B}, \end{array} \right. \end{equation*} where $Δ_{\mathbb{B}}$ is Fuchsian type Laplace operator investigated with totally characteristic degeneracy on the boundary $x_1 =0$, and the nonlinearity $f$ has the subcritical exponential growth or the critical exponential growth.

math.AP↗