CPAA
Infinitely many sign-changing solutions for the Brézis-Nirenberg problem
Jijiang Sun Shiwang Ma
Communications on Pure & Applied Analysis 2014, 13(6): 2317-2330 doi: 10.3934/cpaa.2014.13.2317
In this paper, we present a new proof on the existence of infinitely many sign-changing solutions for the following Brézis-Nirenberg problem \begin{eqnarray} -\Delta u=\lambda u+|u|^{2^{*}-2}u \quad \textrm{in}\, \Omega, \qquad u=0 \quad \textrm{on}\,\partial\Omega, \end{eqnarray} for each fixed $\lambda>0$, under the assumptions that $N\geq 7$, where $\Omega$ is a bounded smooth domain of $\mathbb{R}^{N}$, $2^{*}=\frac{2N}{N-2}$ is the critical Sobolev exponent. In order to construct sign-changing solutions, we will use a combination of invariant sets method and Ljusternik-Schnirelman type minimax method, which is much simpler than the proof of [20] depending on the estimates of Morse indices of nodal solutions to obtain the same result.
keywords: minimax method. sign-changing solutions Critical exponent invariant sets
DCDS-B
Some existence results on periodic and subharmonic solutions of ordinary $P$-Laplacian systems
Yuxiang Zhang Shiwang Ma
Discrete & Continuous Dynamical Systems - B 2009, 12(1): 251-260 doi: 10.3934/dcdsb.2009.12.251
Some existence theorems are obtained for periodic and subharmonic solutions of ordinary $P$-Laplacian systems by the minimax methods in critical point theory.
keywords: Periodic solution; subharmonic solution; $p$-Laplacian system; subquadratic condition; Minimax methods.
CPAA
Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance
Shiwang Ma
Communications on Pure & Applied Analysis 2013, 12(6): 2361-2380 doi: 10.3934/cpaa.2013.12.2361
In this paper, we consider the existence of nontrivial $1$-periodic solutions of the following Hamiltonian systems \begin{eqnarray} -J\dot{z}=H'(t,z), z\in R^{2N}, \end{eqnarray} where $J$ is the standard symplectic matrix of $2N\times 2N$, $H\in C^2 ( [0,1] \times R^{2N}, R)$ is $1$-periodic in its first variable and $H'(t,z)$ denotes the gradient of $H$ with respect to the variable $z$. Furthermore, $H'(t,z)$ is asymptotically linear both at origin and at infinity. Based on the precise computations of the critical groups, Maslov-type index theory and Galerkin approximation procedure, we obtain some existence results for nontrivial $1$-periodic solutions under new classes of conditions. It turns out that our main results improve sharply some known results in the literature.
keywords: periodic solution Critical group resonance. Galerkin approximation Maslov-type index
DCDS
Global asymptotic stability of minimal fronts in monostable lattice equations
Shiwang Ma Xiao-Qiang Zhao
Discrete & Continuous Dynamical Systems - A 2008, 21(1): 259-275 doi: 10.3934/dcds.2008.21.259
The global asymptotic stability with phase shift of traveling wave fronts of minimal speed, in short minimal fronts, is established for a large class of monostable lattice equations via the method of upper and lower solutions and a squeezing technique.
keywords: minimal speed lattice equations. Asymptotic stability traveling waves
CPAA
Nontrivial solutions for Kirchhoff type equations via Morse theory
Jijiang Sun Shiwang Ma
Communications on Pure & Applied Analysis 2014, 13(2): 483-494 doi: 10.3934/cpaa.2014.13.483
In this paper, the existence of nontrivial solutions is obtained for a class of Kirchhoff type problems with Dirichlet boundary conditions by computing the critical groups and Morse theory.
keywords: critical groups Morse theory local linking. Kirchhoff type problems
DCDS-B
Unboundedness of solutions for perturbed asymmetric oscillators
Lixia Wang Shiwang Ma
Discrete & Continuous Dynamical Systems - B 2011, 16(1): 409-421 doi: 10.3934/dcdsb.2011.16.409
In this paper, we consider the existence of unbounded solutions and periodic solutions for the perturbed asymmetric oscillator with damping

$x'' + f(x )x' + ax^+ - bx^-$ $+ g(x)=p(t), $

where $x^+ =\max\{x,0\}, x^-$ $=\max\{-x,0\}$, $a$ and $b$ are two positive constants, $f(x)$ is a continuous function and $ p(t)$ is a $2\pi $-periodic continuous function, $g(x)$ is locally Lipschitz continuous and bounded. We discuss the existence of periodic solutions and unbounded solutions under two classes of conditions: the resonance case $\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}}\in Q$ and the nonresonance case $\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}} \notin Q$. Unlike many existing results in the literature where the function $g(x)$ is required to have asymptotic limits at infinity, our main results here allow $g(x)$ be oscillatory without asymptotic limits.

keywords: resonance periodic solution Unbounded solution nonresonance.
DCDS-B
Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent
Jing Zhang Shiwang Ma
Discrete & Continuous Dynamical Systems - B 2016, 21(6): 1999-2009 doi: 10.3934/dcdsb.2016033
In this paper, we are concerned with the following nonlinear Schrödinger equations with hardy potential and critical Sobolev exponent \begin{equation}\label{eq0.1} \left\{\begin{array}{ll} -\Delta u+\lambda a(x)u^q=\mu\frac{u}{|x|^{2}}+|u|^{2^{*}-2}u,& \textrm{in}\, \mathbb{R}^N, \\ u>0, & \textrm{in}\,\mathcal{D}^{1,2}(\mathbb{R}^N), (1) \end{array} \right. \end{equation} where $2^{*}=\frac{2N}{N-2}$ is the critical Sobolev exponent, $0\leq \mu<\overline{\mu}=\frac{(N-2)^2}{4}$, $a(x)\in C(\mathbb{R}^N)$. We first use an abstract perturbation method in critical point theory to obtain the existence of positive solutions of (1) for small value of $|\lambda|$. Secondly, we focus on an anisotropic elliptic equation of the form \begin{equation}\label{eq0.2} -{\rm div}(B_\lambda(x)\nabla u)+\lambda a(x)u^q=\mu\frac{u}{|x|^{2}}+|u|^{2^*-2}u, x\in\mathbb{R}^N. (2) \end{equation} The same abstract method is used to yield existence result of positive solutions of (2) for small value of $|\lambda|$.
keywords: Positive solutions anisotropic problem. critical exponent pertubation method
DCDS
Bound state solutions of Schrödinger-Poisson system with critical exponent
Xu Zhang Shiwang Ma Qilin Xie
Discrete & Continuous Dynamical Systems - A 2017, 37(1): 605-625 doi: 10.3934/dcds.2017025
In this paper, we consider the following Schrödinger-Poisson problem
$\tag{P}\label{0.1} \begin{cases}- Δ u+V(x)u+K(x)φ u=|u|^{2^*-2}u, &x∈ \mathbb{R}^3,\\-Δ φ=K(x)u^2,&x∈ \mathbb{R}^3,\end{cases}$
where
$2^*=6 $
is the critical exponent in
$\mathbb R^3$
,
$ K∈ L^{\frac{1}{2}}(\mathbb{R}^3)$
and
$V∈ L^{\frac{3}{2}}(\mathbb{R}^3)$
are given nonnegative functions. When
$|V|_{\frac{3}{2}}+|K|_{\frac{1}{2}}$
is suitable small, we prove that problem (P) has at least one bound state solution via a linking theorem.
keywords: Schrödinger-Poisson system critical exponent bound state solutions

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