arXiv subjects

Binlin Zhang

Publications and source records attributed to Binlin Zhang.

10 recordsLinked to original sources

arXiv2021

Normalized ground states for the critical fractional NLS equation with a perturbation

In this paper, we study normalized ground states for the following critical fractional NLS equation with prescribed mass: \begin{equation*} \begin{cases} (-Δ)^{s}u=λu +μ|u|^{q-2}u+|u|^{2_{s}^{\ast}-2}u,&x\in\mathbb{R}^{N}, \int_{\mathbb{R}^{N}}u^{2}dx=a^{2},\\ \end{cases} \end{equation*} where $(-Δ)^{s}$ is the fractional Laplacian, $0 2s$, $2 0$, $μ\in \mathbb{R}$. By using Jeanjean's trick in \cite{Jeanjean}, and the standard method which can be found in \cite{Brezis} to overcome the lack of compactness, we first prove several existence and nonexistence results for a $L^{2}$-subcritical (or $L^{2}$-critical or $L^{2}$-supercritical) perturbation $μ|u|^{q-2}u$, then we give some results about the behavior of the ground state obtained above as $μ\rightarrow 0^{+}$. Our results extend and improve the existing ones in several directions.

math.AP↗

arXiv2020

Positive supersolutions for the Lane-Emden system with inverse-square potentials

In this paper, we study the nonexistence of positive supersolutions for the following Lane-Emden system with inverse-square potentials \begin{equation}\label{0} \left\{ \begin{array}{lll} -Δu+\frac{μ_1}{|x|^2} u= v^p \quad {\rm in}\ \, Ω\setminus\{0\},\\[2mm] -Δv+\frac{μ_2}{|x|^2} v= u^q \quad {\rm in}\ \, Ω\setminus\{0\} \end{array} \right. \end{equation} for suitable $p,q>0$, $μ_1,μ_2\geq -(N-2)^2/4$, where $Ω$ is a smooth bounded domain containing the origin in $\mathbb{R}^N$ with $N\geq 3$. Precisely, we provide sharp supercritical regions of $(p,q)$ for the nonexistence of positive supersolutions to system (\ref{0}) in the cases $-(N-2)^2/4\leq μ_1,μ_2<0$ and $-(N-2)^2/4\leq μ_1<0\leq μ_2$. Due to the negative coefficients $μ_1,μ_2$ of the inverse-square potentials, an initial blowing-up at the origin could be derived and an iteration procedure could be applied in the supercritical case to improve the blowing-up rate until the nonlinearities are not admissible in some weighted $L^1$ spaces. In the subcritical case, we prove the existence of positive supersolutions for system (\ref{s 1.1}) by specific radially symmetric functions.

math.AP↗

arXiv2020

Fractional magnetic Schrödinger-Kirchhoff problems with convolution and critical nonlinearities

In this paper we are concerned with the existence and multiplicity of solutions for the fractional Choquard-type Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity: \begin{eqnarray*} \begin{cases} \varepsilon^{2s}M([u]_{s,A}^2)(-Δ)_{A}^su + V(x)u = (|x|^{-α}*F(|u|^2))f(|u|^2)u + |u|^{2_s^\ast-2}u,\ \ \ x\in \mathbb{R}^N,\\ u(x) \rightarrow 0,\ \ \quad \mbox{as}\ |x| \rightarrow \infty, \end{cases} \end{eqnarray*} where $(-Δ)_{A}^s$ is the fractional magnetic operator with $0 0$ is a positive parameter. The electric potential $V\in C(\mathbb{R}^N, \mathbb{R}^+_0)$ satisfies $V(x) = 0$ in some region of $\mathbb{R}^N$, which means that this is the critical frequency case. We first prove the $(PS)_c$ condition, by using the fractional version of the concentration compactness principle. Then, applying also the mountain pass theorem and the genus theory, we obtain the existence and multiplicity of semiclassical states for the above problem. The main feature of our problems is that the Kirchhoff term $M$ can vanish at zero.

math.AP↗

arXiv2020

Quasi-periodic relativistic strings in the Minkowski space $\textbf{R}^{1+n}$

In this article we consider the motion of relativistic strings in the Minkowski space $\textbf{R}^{1+n}$. Those surfaces are known as a timelike minimal surface, and described by a system with $n$ nonlinear wave equations of Born-Infeld type. By constructing a suitable Nash-Moser iteration scheme, we prove that the $n$-dimensional relativistic strings can admit a more generalized time quasi-periodic motion in $\textbf{R}^{1+n}$. Moreover, those time quasi-periodic solutions are also timelike solutions.

math.DS↗

arXiv2019

Long time existence for the bosonic membrane in the light cone gauge

This paper mainly aims to establish the well-posedness on time interval $[0,\varepsilon^{-\frac{1}{2}}T]$ of the classical initial problem for the bosonic membrane in the light cone gauge. Here $\varepsilon$ is the small parameter measures the nonlinear effects. In geometric, the bosonic membrane are timelike submanifolds with vanishing mean curvature. Since the initial Riemannian metric may be degenerate or non-degenerate, the corresponding equation can be reduce to a quasi-linear degenerate or non-degenerate hyperbolic system of second order with an area preserving constraint via a Hamiltonian reduction.

math.DG↗

arXiv2019

Existence and multiplicity of solutions for fractional Schrödinger-Kirchhoff equations with Trudinger-Moser nonlinearity

We study the existence and multiplicity of solutions for a class of fractional Schrödinger-Kirchhoff type equations with the Trudinger-Moser nonlinearity. More precisely, we consider \begin{gather*} \begin{cases} M\big(\|u\|^{N/s}\big)\left[(-Δ)^s_{N/s}u+V(x)|u|^{\frac{N}{s}-1}u\right]= f(x,u) +λh(x)|u|^{p-2}u\, &{\rm in}\ \ \mathbb{R}^N,\\ \|u\|=\left(\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{N/s}}{|x-y|^{2N}}dxdy+\int_{\mathbb{R}^N}V(x)|u|^{N/s}dx\right)^{s/N}, \end{cases}\end{gather*} where $M:[0,\infty]\rightarrow [0,\infty)$ is a continuous function, $s\in (0,1)$, $N\geq2$, $λ>0$ is a parameter, $1<p<\infty$, $(-Δ)^s_{N/s}$ is the fractional $N/s$--Laplacian, $V:\mathbb{R}^N\rightarrow(0,\infty)$ is a continuous function, $f:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R} $ is a continuous function, and $h:\mathbb{R}^N\rightarrow[0,\infty)$ is a measurable function. First, using the mountain pass theorem, a nonnegative solution is obtained when $f$ satisfies exponential growth conditions and $λ$ is large enough, and we prove that the solution converges to zero in $W_V^{s,N/s}(\mathbb{R}^N)$ as $λ\rightarrow\infty$. Then, using the Ekeland variational principle, a nonnegative nontrivial solution is obtained when $λ$ is small enough, and we show that the solution converges to zero in $W_V^{s,N/s}(\mathbb{R}^N)$ as $λ\rightarrow0$. Furthermore, using the genus theory, infinitely many solutions are obtained when $M$ is a special function and $λ$ is small enough. We note that our paper covers a novel feature of Kirchhoff problems, that is, the Kirchhoff function $M(0)=0$.

math.AP↗

arXiv2018

On the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity

We consider the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity $\varepsilon^{2s}M([u]_{s,A_\varepsilon}^2)(-Δ)_{A_\varepsilon}^su + V(x)u =$ $|u|^{2_s^\ast-2}u + h(x,|u|^2)u,$ $\ \ x\in \mathbb{R}^N,$ where $ u(x) \rightarrow 0$ as $|x| \rightarrow \infty,$ and $(-Δ)_{A_\varepsilon}^s$ is the fractional magnetic operator with $0<s<1$, $2_s^\ast = 2N/(N-2s),$ $M : \mathbb{R}^{+}_{0} \rightarrow \mathbb{R}^{+}$ is a continuous nondecreasing function, $V:\mathbb{R}^N \rightarrow \mathbb{R}^+_0,$ and $A: \mathbb{R}^N \rightarrow \mathbb{R}^N$ are the electric and the magnetic potential, respectively. By using the fractional version of the concentration compactness principle and variational methods, we show that the above problem: (i) has at least one solution provided that $\varepsilon < \mathcal {E}$; and (ii) for any $m^\ast \in \mathbb{N}$, has $m^\ast$ pairs of solutions if $\varepsilon < \mathcal {E}_{m^\ast}$, where $\mathcal {E}$ and $\mathcal {E}_{m^\ast}$ are sufficiently small positive numbers. Moreover, these solutions $u_\varepsilon \rightarrow 0$ as $\varepsilon \rightarrow 0$.

math.AP↗

arXiv2017

On isolated singularities of Kirchhoff--type Laplacian problems

In this paper, we study isolated singular positive solutions for the following Kirchhoff--type Laplacian problem: \begin{equation*} -\left(θ+\int_Ω |\nabla u| dx\right)Δu =u^p \quad{\rm in}\quad Ω\setminus \{0\},\qquad u=0\quad {\rm on}\quad \partial Ω, \end{equation*} where $p>1$, $θ\in \R$, $Ω$ is a bounded smooth domain containing the origin in $\R^N$ with $N\ge 2$. In the subcritical case: $1 0$. To estimate $M_θ(u)$, we make use of the rearrangement argument. Furthermore, we obtain a sequence of isolated singular solutions such that $M_θ(u)<0$, by analyzing relationship between the parameter $λ$ and the unique solution $u_λ$ of $$-Δu+λu^p=kδ_0\quad{\rm in}\quad B_1(0),\qquad u=0\quad {\rm on}\quad \partial B_1(0).$$ In the supercritical case: $N/(N-2)\le p<(N+2)/(N-2)$ with $N\ge3$, we obtain two isolated singular solutions $u_i$ with $i=1,2$ such that $M_θ(u_i)>0$ under some appropriate assumptions.

math.AP↗

arXiv2017

Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials

This paper is concerned with the following fractional Schrödinger equations involving critical exponents: \begin{eqnarray*} (-Δ)^αu+V(x)u=k(x)f(u)+λ|u|^{2_α^{*}-2}u\quad\quad \mbox{in}\ \mathbb{R}^{N}, \end{eqnarray*} where $(-Δ)^α$ is the fractional Laplacian operator with $α\in(0,1)$, $N\geq2$, $λ$ is a positive real parameter and $2_α^{*}=2N/(N-2α)$ is the critical Sobolev exponent, $V(x)$ and $k(x)$ are positive and bounded functions satisfying some extra hypotheses. Based on the principle of concentration compactness in the fractional Sobolev space and the minimax arguments, we obtain the existence of a nontrivial radially symmetric weak solution for the above-mentioned equations without assuming the Ambrosetti-Rabinowitz condition on the subcritical nonlinearity.

math.AP↗

arXiv2016

Nonlocal Schrödinger-Kirchhoff equations with external magnetic field

The paper deals with existence and multiplicity of solutions of the fractional Schrödinger--Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case \begin{equation*} (a+b[u]_{s,A}^{2θ-2})(-Δ)_A^su+V(x)u=f(x,|u|)u\,\, \quad \text{in $\mathbb{R}^N$}, \end{equation*} where $s\in (0,1)$, $N>2s$, $a\in \mathbb{R}^+_0$, $b\in \mathbb{R}^+_0$, $θ\in[1,N/(N-2s))$, $A:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a magnetic potential, $V:\mathbb{R}^N\rightarrow \mathbb{R}^+$ is an electric potential, $(-Δ)_A^s$ is the fractional magnetic operator. In the super- and sub-linear cases, the existence of least energy solutions for the above problem is obtained by the mountain pass theorem, combined with the Nehari method, and by the direct methods respectively. In the superlinear-sublinear case, the existence of infinitely many solutions is investigated by the symmetric mountain pass theorem.

math.AP↗