Infinitely many sign-changing solutions for the Brézis-Nirenberg problem
Jijiang Sun Shiwang Ma
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
Some existence results on periodic and subharmonic solutions of ordinary $P$-Laplacian systems
Yuxiang Zhang Shiwang Ma
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.
Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance
Shiwang Ma
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
Global asymptotic stability of minimal fronts in monostable lattice equations
Shiwang Ma Xiao-Qiang Zhao
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
Nontrivial solutions for Kirchhoff type equations via Morse theory
Jijiang Sun Shiwang Ma
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
Unboundedness of solutions for perturbed asymmetric oscillators
Lixia Wang Shiwang Ma
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.
Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent
Jing Zhang Shiwang Ma
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
Bound state solutions of Schrödinger-Poisson system with critical exponent
Xu Zhang Shiwang Ma Qilin Xie
In this paper, we consider the following Schrödinger-Poisson problem
$\tag{P}\label{0.1} \begin{cases}- Δ u+V(x)u+K(x)φ u=|u|^{2^*-2}u, &x∈ \mathbb{R}^3,\\-Δ φ=K(x)u^2,&x∈ \mathbb{R}^3,\end{cases}$
$2^*=6 $
is the critical exponent in
$\mathbb R^3$
$ K∈ L^{\frac{1}{2}}(\mathbb{R}^3)$
$V∈ L^{\frac{3}{2}}(\mathbb{R}^3)$
are given nonnegative functions. When
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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