Periodic solutions of Hamiltonian systems with anisotropic growth
Tianqing An Zhi-Qiang Wang
Communications on Pure & Applied Analysis 2010, 9(4): 1069-1082 doi: 10.3934/cpaa.2010.9.1069
In this paper we obtain some existence and multiplicity results for periodic solutions of nonautonomous Hamiltonian systems $\dot z(t)=J\nabla H(z(t),t)$ whose Hamiltonian functions may have simultaneously, in different components, superquadratic, subquadratic and quadratic behaviors. Our results generalize some earlier work [3] of P. Felmer and [5] of P. Felmer and Z.-Q. Wang.
keywords: growth conditions Hamiltonian system minimax theorems. Periodic solution
Positive solutions to a class of quasilinear elliptic equations on $\mathbb R$
Antonio Ambrosetti Zhi-Qiang Wang
Discrete & Continuous Dynamical Systems - A 2003, 9(1): 55-68 doi: 10.3934/dcds.2003.9.55
We discuss the existence of positive solutions of perturbation to a class of quasilinear elliptic equations on $\mathbb R$.
keywords: perturbation theory. Fully nonlinear elliptic equations
A complete classification of ground-states for a coupled nonlinear Schrödinger system
Chuangye Liu Zhi-Qiang Wang
Communications on Pure & Applied Analysis 2017, 16(1): 115-130 doi: 10.3934/cpaa.2017005
In this paper, we establish the existence of nontrivial ground-state solutions for a coupled nonlinear Schrödinger system
$-\Delta u_j+ u_j=\sum\limits_{i=1}^mb_{ij}u_i^2u_j, \quad\text{in}\ \mathbb{R}^n,\\ u_j(x)\to 0\ \text{as}\ |x|\ \to \infty, \quad j=1,2,\cdots, m,$
where $n=1, 2, 3, m\geq 2$ and $b_{ij}$ are positive constants satisfying $b_{ij}=b_{ji}.$ By nontrivial we mean a solution that has all components non-zero. Due to possible systems collapsing it is important to classify ground state solutions. For $m=3$, we get a complete picture that describes whether nontrivial ground-state solutions exist or not for all possible cases according to some algebraic conditions of the matrix $B = (b_{ij})$. In particular, there is a nontrivial ground-state solution provided that all coupling constants $b_{ij}, i\neq j$ are sufficiently large as opposed to cases in which any ground-state solution has at least a zero component when $b_{ij}, i\neq j$ are all sufficiently small. Moreover, we prove that any ground-state solution is synchronized when matrix $B=(b_{ij})$ is positive semi-definite.
keywords: Ground states synchronized solutions coupled Schrödinger systems
On the least energy sign-changing solutions for a nonlinear elliptic system
Yohei Sato Zhi-Qiang Wang
Discrete & Continuous Dynamical Systems - A 2015, 35(5): 2151-2164 doi: 10.3934/dcds.2015.35.2151
In this paper, as bound state solutions we consider least energy sign-changing solutions to a nonlinear elliptic system which consists of N-equations defined on a bounded domain $\Omega$. For any subset $K\subset \{1,\cdots, N\}$, we show the existence of sign-changing solution $\vec{u}=(u_1,\cdots,u_n)$ such that, for $i\in K$, $u_i$ are sign-changing functions that change sign exactly once in $\Omega$, and, for $i\notin K$, $u_i$ are one sign functions. We give a variational characterization of such solutions on modified Nehari type constrained sets.
keywords: sign-changing solutions least energy solution multiple existence of solutions Nonlinear Schrödinger system
On a class of anisotropic nonlinear elliptic equations in $\mathbb R^N$
Francois van Heerden Zhi-Qiang Wang
Communications on Pure & Applied Analysis 2008, 7(1): 149-162 doi: 10.3934/cpaa.2008.7.149
We establish some imbedding results of weighted Sobolev spaces. The results then are used to obtain ground state solutions of nonlinear elliptic equations with anisotropic coefficients.
keywords: weighted Sobolev spaces ground states. Imbedding
Bifurcation results on positive solutions of an indefinite nonlinear elliptic system
Rushun Tian Zhi-Qiang Wang
Discrete & Continuous Dynamical Systems - A 2013, 33(1): 335-344 doi: 10.3934/dcds.2013.33.335
Consider the following nonlinear elliptic system \begin{equation*} \left\{\begin{array}{ll} -\Delta u - u=\mu_1u^3+\beta uv^2,\ & \hbox{in}\ \Omega\\ -\Delta v - v= \mu_2v^3+\beta vu^2,\ & \hbox{in}\ \Omega\\ u,v>0\ \hbox{in}\ \Omega, \ u=v=0,\ & \hbox{on}\ \partial\Omega, \end{array} \right. \end{equation*}where $\mu_1,\mu_2>0$ are constants and $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ for $N\leq3$. We study the existence and non-existence of positive solutions and give bifurcation results in terms of the coupling constant $\beta$.
keywords: indefinite system. Positive solutions bifurcations
Subharmonic solutions for second order Hamiltonian systems
Norimichi Hirano Zhi-Qiang Wang
Discrete & Continuous Dynamical Systems - A 1998, 4(3): 467-474 doi: 10.3934/dcds.1998.4.467
In this paper, we are interested in the existence of subharmonic solutions for the problem $ u_{t t} + G'(u) = f(t), $ where $G:R^{ N} \rightarrow R$ is not necessarily convex and $f:R \rightarrow R^N$ is periodic with minimal period $T > 0$.
keywords: subharmonic solutions. Second order Hamiltonian systems
Synchronization of positive solutions for coupled Schrödinger equations
Chuangye Liu Zhi-Qiang Wang
Discrete & Continuous Dynamical Systems - A 2018, 38(6): 2795-2808 doi: 10.3934/dcds.2018118
In this paper, we analyze synchronized positive solutions for a coupled nonlinear Schrödinger equation
$\left\{ {\begin{array}{*{20}{c}} {\Delta u - u + ({\mu _1}|u{|^p} + \beta |v{|^p})|u{|^{p - 2}}u = 0,}&{{\text{i}}n\;{\mathbb{R}^n},} \\ {\Delta v - v + ({\mu _2}|v{|^p} + \beta |u{|^p})|v{|^{p - 2}}v = 0,}&{{\text{i}}n\;{\mathbb{R}^n},} \end{array}} \right.$
$ 2< p<\frac{n}{n-2}, $
$ n\ge 3$
$ 2< p<+∞ $
, if
$ n = 1, 2, $
$μ_1, μ_2, β>0 $
are positive constants. Our goal is two fold. On one hand we study the question under what conditions the ground states are nontrivial synchronized positive solutions, giving precise conditions in terms of the size of the coupling constant. On the other hand, we examine the questions on whether all positive solutions are synchronized solutions. We have a complete answer for the case
$ n = 1 $
by proving that positivity implies synchronization. The latter result enables us to obtain the exact number of positive solutions even though no uniqueness result holds in the case, and this is quite different from the case
$ p = 2 $
for which uniqueness of positive solutions was known ([19]).
keywords: Ground-states coupled systems synchronization
Asymptotic uniqueness and exact symmetry of k-bump solutions for a class of degenerate elliptic problems
Florin Catrina Zhi-Qiang Wang
Conference Publications 2001, 2001(Special): 80-87 doi: 10.3934/proc.2001.2001.80
Please refer to Full Text.
On a new index theory and non semi-trivial solutions for elliptic systems
Kung-Ching Chang Zhi-Qiang Wang Tan Zhang
Discrete & Continuous Dynamical Systems - A 2010, 28(2): 809-826 doi: 10.3934/dcds.2010.28.809
Two indices, which are similar to the Krasnoselski's genus on the sphere, are defined on the product of spheres. They are applied to investigate the multiple non semi-trivial solutions for elliptic systems. Both constraint and unconstraint problems are studied.
keywords: genus. Elliptic system eigenvalue critical point theory

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