CPAA
Disconjugacy and extremal solutions of nonlinear third-order equations
Ruyun Ma Yanqiong Lu
Communications on Pure & Applied Analysis 2014, 13(3): 1223-1236 doi: 10.3934/cpaa.2014.13.1223
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 $u'''+Mu''+f(t,u)=0$, $u(0)=u'(0)=u(1)=0$.
keywords: lower and upper solutions. disconjugacy Third-order equations positive solutions
DCDS
Unilateral global bifurcation for $p$-Laplacian with non-$p-$1-linearization nonlinearity
Guowei Dai Ruyun Ma
Discrete & Continuous Dynamical Systems - A 2015, 35(1): 99-116 doi: 10.3934/dcds.2015.35.99
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.
keywords: Unilateral bifurcation nodal solutions half-quasilinear problems $p$-Laplacian.
DCDS-B
Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space
Ruyun Ma Man Xu
Discrete & Continuous Dynamical Systems - B 2017, 22(11): 1-18 doi: 10.3934/dcdsb.2018271
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
$Ω$
is a bounded regular domain in
$\mathbb{R}^N$
, the function
$f$
satisfies the Carathéodory conditions, and
$f$
is either superlinear or sublinear in
$u$
at
$0$
. The proof of our main results are based upon bifurcation techniques.
keywords: Prescribed mean curvature equations Lorentz-Minkowski space positive solution bifurcation connected component
DCDS-B
Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal
Guo-Bao Zhang Ruyun Ma Xue-Shi Li
Discrete & Continuous Dynamical Systems - B 2018, 23(2): 587-608 doi: 10.3934/dcdsb.2018035

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
Eigenvalues, bifurcation and one-sign solutions for the periodic $p$-Laplacian
Guowei Dai Ruyun Ma Haiyan Wang
Communications on Pure & Applied Analysis 2013, 12(6): 2839-2872 doi: 10.3934/cpaa.2013.12.2839
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.
keywords: One-sign solutions. Eigenvalues Unilateral global bifurcation Periodic $p$-Laplacian

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