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### Open Access Journals

DCDS

We discuss the existence of positive
solutions of perturbation to a class of quasilinear elliptic
equations on $\mathbb R$.

DCDS

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.

CPAA

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.

DCDS

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 $\beta$.

DCDS

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 $T > 0$.

PROC

Please refer to Full Text.

keywords:

CPAA

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.

CPAA

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.

DCDS

In this paper, we analyze synchronized positive solutions for a coupled nonlinear Schrödinger equation

$\left\{ {\begin{array}{*{20}{c}} {\Delta u - u + ({\mu _1}|u{|^p} + \beta |v{|^p})|u{|^{p - 2}}u = 0,}&{{\text{i}}n\;{\mathbb{R}^n},} \\ {\Delta v - v + ({\mu _2}|v{|^p} + \beta |u{|^p})|v{|^{p - 2}}v = 0,}&{{\text{i}}n\;{\mathbb{R}^n},} \end{array}} \right.$ |

where

if

and

, if

and

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

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

for which uniqueness of positive solutions was known ([19 ]).

$ 2< p<\frac{n}{n-2}, $ |

$ n\ge 3$ |

$ 2< p<+∞ $ |

$ n = 1, 2, $ |

$μ_1, μ_2, β>0 $ |

$ n = 1 $ |

$ p = 2 $ |

DCDS

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.

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