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### Open Access Journals

CPAA

In this paper, we make an exhaustive
study of the third order linear operators $u''' +Mu$, $u'''+Mu'$ and
$u'''+Mu''$ coupled with $(k, 3-k)$-conjugate boundary conditions,
where $k=1,2$. We obtain the optimal intervals on which the Green's functions are of one sign. The main tool is the disconjugacy
theory. As an application of our results, we develop a monotone
iteration method to obtain positive solutions of the nonlinear
problem $u'''+Mu''+f(t,u)=0$, $u(0)=u'(0)=u(1)=0$.

DCDS

In this paper, we establish a unilateral global bifurcation result from interval for a class of $p$-Laplacian problems.
By applying above result, we study the spectrum of a class of half-quasilinear problems.
Moreover, we also investigate the existence of nodal solutions for a class of half-quasilinear eigenvalue problems.

DCDS-B

In this paper we study global bifurcation phenomena for the Dirichlet problem associated with the prescribed mean curvature equation in Minkowski space

$\left\{ \begin{array}{l} -\text{div}\big(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\big) = λ f(x,u,\nabla u)\ \ \ \ \ \ & \text{in}\ Ω,\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u = 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & \text{on}\ \partial Ω.\\\end{array} \right.$ |

Here

is a bounded regular domain in

, the function

satisfies the Carathéodory conditions, and

is either superlinear or sublinear in

at

. The proof of our main results are based upon bifurcation techniques.

$Ω$ |

$\mathbb{R}^N$ |

$f$ |

$f$ |

$u$ |

$0$ |

DCDS-B

This paper is concerned with the traveling waves of a nonlocal dispersal Lotka-Volterra strong competition model with bistable nonlinearity. We first establish the asymptotic behavior of traveling waves at infinity. Then by applying the stronger comparison principle and the sliding method, we prove that the traveling waves with nonzero speed are strictly monotone. Moreover, the uniqueness of wave speeds is also obtained.

keywords:
Nonlocal dispersal
,
strong competition system
,
traveling waves
,
monotonicity
,
uniqueness
,
wave speed

CPAA

In this paper, we establish a unilateral global bifurcation result for a class of quasilinear periodic boundary problems with a sign-changing weight.
By the Ljusternik-Schnirelmann theory, we first study the spectrum of the periodic $p$-Laplacian with the sign-changing weight.
In particular, we show that there exist two simple, isolated, principal eigenvalues $\lambda_0^+$ and $\lambda_0^-$.
Furthermore, under some natural hypotheses on perturbation function,
we show that $(\lambda_0^\nu,0)$ is a bifurcation
point of the above problems and there are two distinct unbounded sub-continua
$C_\nu^{+}$ and $C_\nu^{-}$,
consisting of the continuum $C_\nu$ emanating from $(\lambda_0^\nu, 0)$,
where $\nu\in\{+,-\}$. As an application of the above result, we study the existence of one-sign solutions
for a class of quasilinear periodic boundary problems with the sign-changing weight.
Moreover, the uniqueness of one-sign solutions and the dependence of solutions on the parameter $\lambda$ are
also studied.

## Year of publication

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