Positive solutions to a class of quasilinear elliptic equations on $\mathbb R$
Antonio Ambrosetti Zhi-Qiang Wang
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
On the least energy sign-changing solutions for a nonlinear elliptic system
Yohei Sato Zhi-Qiang Wang
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
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
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
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
Asymptotic uniqueness and exact symmetry of k-bump solutions for a class of degenerate elliptic problems
Florin Catrina Zhi-Qiang Wang
Please refer to Full Text.
Periodic solutions of Hamiltonian systems with anisotropic growth
Tianqing An Zhi-Qiang Wang
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
A complete classification of ground-states for a coupled nonlinear Schrödinger system
Chuangye Liu Zhi-Qiang Wang
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 a new index theory and non semi-trivial solutions for elliptic systems
Kung-Ching Chang Zhi-Qiang Wang Tan Zhang
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
On electro-kinetic fluids: One dimensional configurations
R. Ryham Chun Liu Zhi-Qiang Wang
Electro-kinetic fluids can be modeled by hydrodynamic systems describing the coupling between fluids and electric charges. The system consists of a momentum equation together with transport equations of charges. In the dynamics, the special coupling between the Lorentz force in the velocity equation and the material transport in the charge equation gives an energy dissipation law. In stationary situations, the system reduces to a Poisson-Boltzmann type of equation. In particular, under the no flux boundary conditions, the conservation of the total charge densities gives nonlocal integral terms in the equation. In this paper, we analyze the qualitative properties of solutions to such an equation, especially when the Debye constant $\epsilon$ approaches zero. Explicit properties can be derived for the one dimensional case while some may be generalized to higher dimensions. We also present some numerical simulation results of the system.
keywords: Nernst-Plank. electrokinetics Poisson-Boltzmann Debye length

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