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.
CPAA
Multiple solutions for a fractional nonlinear Schrödinger equation with local potential
Wulong Liu Guowei Dai
Communications on Pure & Applied Analysis 2017, 16(6): 2105-2123 doi: 10.3934/cpaa.2017104
Using penalization techniques and the Ljusternik-Schnirelmann theory, we establish the multiplicity and concentration of solutions for the following fractional Schrödinger equation
$\left\{ \begin{align} &{{\varepsilon }^{2\alpha }}{{\left( -\Delta \right)}^{a}}u+V\left( x \right)u=f\left( u \right),\ \ x\in {{\mathbb{R}}^{N}}, \\ &u\in {{H}^{a}}\left( {{\mathbb{R}}^{N}} \right),u>0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in {{\mathbb{R}}^{N}}, \\ \end{align} \right.$
where
$0<α<1$
,
$N>2α$
,
$\varepsilon>0$
is a small parameter,
$V$
satisfies the local condition, and
$f$
is superlinear and subcritical nonlinearity. We show that this equation has at least
$\text{cat}_{M_{δ}}(M)$
single spike solutions.
keywords: Penalization techniques Ljusternik-Schnirelmann theory fractional Schrödinger equation multiple solutions single spike solutions
DCDS-S
Stability analysis of a model on varying domain with the Robin boundary condition
Xiaofei Cao Guowei Dai
Discrete & Continuous Dynamical Systems - S 2017, 10(5): 935-942 doi: 10.3934/dcdss.2017048

In this paper we develop a non-autonomous reaction-diffusion model with the Robin boundary conditions to describe insect dispersal on an isotropically varying domain. We investigate the stability of the reaction-diffusion model. The stability results of the model describe either insect survival or vanishing.

keywords: Stability bifurcation varying domain insect dispersal model Robin boundary condition
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
DCDS-B
Partial differential equations with Robin boundary condition in online social networks
Guowei Dai Ruyun Ma Haiyan Wang Feng Wang Kuai Xu
Discrete & Continuous Dynamical Systems - B 2015, 20(6): 1609-1624 doi: 10.3934/dcdsb.2015.20.1609
In recent years, online social networks such as Twitter, have become a major source of information exchange and research on information diffusion in social networks has been accelerated. Partial differential equations are proposed to characterize temporal and spatial patterns of information diffusion over online social networks. The new modeling approach presents a new analytic framework towards quantifying information diffusion through the interplay of structural and topical influences. In this paper we develop a non-autonomous diffusive logistic model with indefinite weight and the Robin boundary condition to describe information diffusion in online social networks. It is validated with a real dataset from an online social network, Digg.com. The simulation shows that the logistic model with the Robin boundary condition is able to more accurately predict the density of influenced users. We study the bifurcation, stability of the diffusive logistic model with heterogeneity in distance. The bifurcation and stability results of the model information describe either information spreading or vanishing in online social networks.
keywords: diffusive logistic model indefinite weight Robin boundary condition. Bifurcation online social networks stability
DCDS
Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros
Guowei Dai
Discrete & Continuous Dynamical Systems - A 2016, 36(10): 5323-5345 doi: 10.3934/dcds.2016034
In this paper, we use bifurcation method to investigate the existence and multiplicity of one-sign solutions of the $p$-Laplacian involving a linear/superlinear nonlinearity with zeros. To do this, we first establish a bifurcation theorem from infinity for nonlinear operator equation with homogeneous operator. To deal with the superlinear case, we establish several topological results involving superior limit.
keywords: superior limit topological method. Bifurcation homogeneous operator one-sign solution $p$-Laplacian

Year of publication

Related Authors

Related Keywords

[Back to Top]