DCDS-B

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 $|\lambda|$.

CPAA

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.

DCDS-B

Some existence theorems are obtained for periodic and subharmonic
solutions of ordinary $P$-Laplacian systems by the minimax methods
in critical point theory.

CPAA

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.

DCDS

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.

CPAA

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.

DCDS-B

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.

DCDS

In this paper, we consider the following Schrödinger-Poisson problem

where

is the critical exponent in

,

and

are given nonnegative functions. When

is suitable small, we prove that problem (P) has at least one bound state solution via a linking theorem.