arXiv subjects

Zhitao Zhang

Publications and source records attributed to Zhitao Zhang.

31 records · Page 2Linked to original sources

arXiv2021

Normalized solutions to Schrödinger systems with linear and nonlinear couplings

In this paper, we study important Schrödinger systems with linear and nonlinear couplings \begin{equation}\label{eq:diricichlet} \begin{cases} -Δu_1-λ_1 u_1=μ_1 |u_1|^{p_1-2}u_1+r_1β|u_1|^{r_1-2}u_1|u_2|^{r_2}+κ(x)u_2~\hbox{in}~\mathbb{R}^N,\\ -Δu_2-λ_2 u_2=μ_2 |u_2|^{p_2-2}u_2+r_2β|u_1|^{r_1}|u_2|^{r_2-2}u_2+κ(x)u_1~ \hbox{in}~\mathbb{R}^N,\\ u_1\in H^1(\mathbb{R}^N), u_2\in H^1(\mathbb{R}^N),\nonumber \end{cases} \end{equation} with the condition $$\int_{\mathbb{R}^N} u_1^2=a_1^2, \int_{\mathbb{R}^N} u_2^2=a_2^2,$$ where $N\geq 2$, $μ_1,μ_2,a_1,a_2>0$, $β\in\mathbb{R}$, $2<p_1,p_2<2^*$, $2<r_1+r_2<2^*$, $κ(x)\in L^{\infty}(\mathbb{R}^N)$ with fixed sign and $λ_1,λ_2$ are Lagrangian multipliers. We use Ekland variational principle to prove this system has a normalized radially symmetric solution for $L^2-$subcritical case when $N\geq 2$, and use minimax method to prove this system has a normalized radially symmetric positive solution for $L^2-$supercritical case when $N=3$, $p_1=p_2=4,\ r_1=r_2=2$.

math.AP↗

arXiv2021

Periodic solutions to Klein-Gordon systems with linear couplings

In this paper, we study the nonlinear Klein-Gordon systems arising from relativistic physics and quantum field theories $$\left\{\begin{array}{lll} u_{tt}- u_{xx} +bu + \varepsilon v + f(t,x,u) =0,\; v_{tt}- v_{xx} +bv + \varepsilon u + g(t,x,v) =0 \end{array}\right. $$ where $u,v$ satisfy the Dirichlet boundary conditions on spatial interval $[0, π]$, $b>0$ and $f$, $g$ are $2π$-periodic in $t$. We are concerned with the existence, regularity and asymptotic behavior of time-periodic solutions to the linearly coupled problem as $\varepsilon$ goes to 0. Firstly, under some superlinear growth and monotonicity assumptions on $f$ and $g$, we obtain the solutions $(u_\varepsilon, v_\varepsilon)$ with time-period $2π$ for the problem as the linear coupling constant $\varepsilon$ is sufficiently small, by constructing critical points of an indefinite functional via variational methods. Secondly, we give precise characterization for the asymptotic behavior of these solutions, and show that as $\varepsilon\rightarrow 0$, $(u_\varepsilon, v_\varepsilon)$ converge to the solutions of the wave equations without the coupling terms. Finally, by careful analysis which are quite different from the elliptic regularity theory, we obtain some interesting results concerning the higher regularity of the periodic solutions.

math.AP↗

arXiv2021

Dynamics of nonlinear hyperbolic equations of Kirchhoff type

In this paper, we study the initial boundary value problem of the important hyperbolic Kirchhoff equation $$u_{tt}-\left(a \int_Ω|\nabla u|^2 \dif x +b\right)Δu = λu+ |u|^{p-1}u ,$$ where $a$, $b>0$, $p>1$, $λ\in \mathbb{R}$ and the initial energy is arbitrarily large. We prove several new theorems on the dynamics such as the boundedness or finite time blow-up of solution under the different range of $a$, $b$, $λ$ and the initial data for the following cases: (i) $1 1/Λ$, (iii) $p=3$, $a \leq 1/Λ$ and $\lam b\lam_1$, (v) $p>3$ and $\lam\leq b\lam_1$, (vi) $p>3$ and $\lam> b\lam_1$, where $\lam_1 = \inf\left\{\|\nabla u\|^2_2 :~ u\in H^1_0(Ω)\ {\rm and}\ \|u\|_2 =1\right\}$, and $Λ= \inf\left\{\|\nabla u\|^4_2 :~ u\in H^1_0(Ω)\ {\rm and}\ \|u\|_4 =1\right\}$. Moreover, we prove the invariance of some stable and unstable sets of the solution for suitable $a$, $b$ and $\lam$, and give the sufficient conditions of initial data to generate a vacuum region of the solution. Due to the nonlocal effect caused by the nonlocal integro-differential term, we show many interesting differences between the blow-up phenomenon of the problem for $a>0$ and $a=0$.

math.AP↗

arXiv2020

Normalized solutions to the Chern-Simons-Schrödinger system

In this paper, we study normalized solutions to the Chern-Simons-Schrödinger system, which is a gauge-covariant nonlinear Schöridnger system with a long-range electromagnetic field, arising in nonrelativistic quantum mechanics theory. The solutions correspond to critical points of the underlying energy functional subject to the $L^2$-norm constraint. Our research covers several aspects. Firstly, in the mass subcritical case, we establish the compactness of any minimizing sequence to the associated global minimization problem. As a by-product of the compactness of any minimizing sequence, the orbital stability of the set of minimizers to the problem is achieved. In addition, we discuss the radial symmetry and uniqueness of minimizer to the problem. Secondly, in the mass critical case, we investigate the existence and nonexistence of normalized solution. Finally, in the mass supercritical case, we prove the existence of ground state and infinitely many radially symmetric solutions. Moreover, the instability of ground states is explored

math.AP↗

arXiv2020

Effective Magnetic Hamiltonian at Finite Temperatures for Rare Earth Chalcogenides

Alkali metal rare-earth chalcogenide $ARECh2$ (A=alkali or monovalent metal, RE=rare earth, Ch=O, S, Se, Te), is a large family of quantum spin liquid (QSL) candidates we discovered recently. Unlike $YbMgGaO4$, most members in the family except for the oxide ones, have relatively small crystalline electric-field (CEF) excitation levels, particularly the first ones. This makes the conventional Curie-Weiss analysis at finite temperatures inapplicable and CEF excitations may play an essential role in understanding the low-energy spin physics. Here we considered an effective magnetic Hamiltonian incorporating CEF excitations and spin-spin interactions, to accurately describe thermodynamics in such a system. By taking $NaYbSe2$ as an example, we were able to analyze magnetic susceptibility, magnetization under pulsed high fields and heat capacity in a systematic and comprehensive way. The analysis allows us to produce accurate anisotropic exchange coupling energies and unambiguously determine a crossover temperature ($\sim$25 K in the case of $NaYbSe2$), below which CEF effects fade away and pure spin-spin interactions stand out. We further validated the effective picture by successfully explaining the anomalous temperature dependence of electron spin resonance (ESR) spectral width. The effective scenario in principle can be generalized to other rare-earth spin systems with small CEF excitations.

cond-mat.str-el↗

arXiv2018

On the Lane-Emden conjecture

We consider the Lane-Emden conjecture which states that there is no non-trivial non-negative solution for the Lane-Emden system whenever the pair of exponents is subcritical. By Sobolev embeddings on $S^{N-1}$ and scale invariance of the solutions, we show this conjecture holds in a new region. Our methods can also be used to prove the Lane-Emden conjecture in space dimension $N\leq 4$, that is to give a different proof of the main result of Souplet in Adv. Math. 2009.

math.AP↗

arXiv2018

Symmetry breaking via Morse index for equations and systems of Hénon-Schrödinger type

We consider the Dirichlet problem for the Schrödinger-Hénon system $$ -Δu + μ_1 u = |x|^α\partial_u F(u,v),\quad \qquad -Δv + μ_2 v = |x|^α\partial_v F(u,v) $$ in the unit ball $Ω\subset \mathbb{R}^N, N\geq 2$, where $α>-1$ is a parameter and $F: \mathbb{R}^2 \to \mathbb{R}$ is a $p$-homogeneous $C^2$-function for some $p>2$ with $F(u,v)>0$ for $(u,v) \not = (0,0)$. We show that, as $α\to \infty$, the Morse index of nontrivial radial solutions of this problem (positive or sign-changing) tends to infinity. This result is new even for the corresponding scalar Hénon equation and extends a previous result by Moreira dos Santos and Pacella for the case $N=2$. In particular, the result implies symmetry breaking for ground state solutions, but also for other solutions obtained by an $α$-independent variational minimax principle.

math.AP↗

arXiv2016

Ground and bound state solutions for a Schrödinger system with linear and nonlinear couplings in $\mathbb{R}^N$

We study the existence of ground and bound state solutions for a system of coupled Schrödinger equations with linear and nonlinear couplings in $\mathbb{R}^N$. By studying the limit system and using concentration compactness arguments, we prove the existence of ground and bound state solutions under suitable assumptions. Our results are new even for the limit system.

math.AP↗

arXiv2016

Asymmetric critical $p$-Laplacian problems

We obtain nontrivial solutions for two types of critical $p$-Laplacian problems with asymmetric nonlinearities in a smooth bounded domain in ${\mathbb R}^N,\, N \ge 2$. For $p < N$, we consider an asymmetric problem involving the critical Sobolev exponent $p^\ast = Np/(N - p)$. In the borderline case $p = N$, we consider an asymmetric critical exponential nonlinearity of the Trudinger-Moser type. In the absence of a suitable direct sum decomposition, we use a linking theorem based on the ${\mathbb Z}_2$-cohomological index to obtain our solutions.

math.AP↗

arXiv2015

Existence and bifurcation of solutions for a double coupled system of Schrodinger equations

Consider the following system of double coupled Schrödinger equations arising from Bose-Einstein condensates etc., \begin{equation*} \left\{\begin{array}{l} -Δu + u =μ_1 u^3 + βuv^2- κv, -Δv + v =μ_2 v^3 + βu^2v- κu, u\neq0, v\neq0\ \hbox{and}\ u, v\in H^1(\R^N), \end{array} \right. \end{equation*}where $μ_1, μ_2$ are positive and fixed, $κ$ and $β$ are linear and nonlinear coupling parameters respectively. We first use critical point theory and Liouville type theorem to prove some existence and nonexistence results on the positive solutions of this system. Then using the positive and non-degenerate solution of the scalar equation $-Δω+ω=ω^3$, $ω\in H_r^1(\R^N)$, we construct a synchronized solution branch to prove that for $β$ in certain range and fixed, there exist a series of bifurcations in product space $\R\times H^1_r(\R^N)\times H^1_r(\R^N)$ with parameter $κ$.

math.AP↗

arXiv2014

On a power-type coupled system of Monge-Ampère equations

We study an elliptic system coupled by Monge-Ampère equations: \begin{center} $\left\{ \begin{array}{ll} det~D^{2}u_{1}={(-u_{2})}^α, & \hbox{in $Ω,$} det~D^{2}u_{2}={(-u_{1})}^β, & \hbox{in $Ω,$} u_{1}<0, u_{2}<0,& \hbox{in $Ω,$} u_{1}=u_{2}=0, & \hbox{on $ \partial Ω,$} \end{array} \right.$ \end{center} here $Ω$~is a smooth, bounded and strictly convex domain in~$\mathbb{R}^{N}$,~$N\geq2,~α>0,~β>0$. When $Ω$ is the unit ball in $\mathbb{R}^{N}$, we use index theory of fixed points for completely continuous operators to get existence, uniqueness results and nonexistence of radial convex solutions under some corresponding assumptions on $α,β$. When $α>0$, $β>0$ and $αβ=N^2$ we also study a corresponding eigenvalue problem in more general domains.

math.AP↗

arXiv2014

Existence and multiplicity of solutions for nonlocal systems with Kirchhoff type

Firstly, we use Nehari manifold and Mountain Pass Lemma to prove an existence result of positive solutions for a class of nonlocal elliptic system with Kirchhoff type. Then a multiplicity result is established by cohomological index of Fadell and Rabinowitz. We also consider the critical case and prove existence of positive least energy solution when the parameter $β$ is sufficiently large.

math.AP↗

arXiv2014

Existence of multiple solutions to a class of nonlinear Schrödinger system with external sources terms

We study a class of nonlinear schrödinger system with external sources terms as perturbations in order to obtain existence of multiple solutions, this system arises from Bose-Einstein condensates etc..As these external sources terms are positive functions and small in some sense, we use Nehari manifold to get the existence of a positive ground state solution and a positive bound state solution.

math.AP↗