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Guowei Dai

Publications and source records attributed to Guowei Dai.

29 records · Page 2Linked to original sources
arXiv2020

Component factors in $K_{1,r}$-free graphs

A graph is said to be $K_{1,r}$-free if it does not contain an induced subgraph isomorphic to $K_{1,r}$. An $\mathcal{F}$-factor is a spanning subgraph $H$ such that each connected component of $H$ is isomorphic to some graph in $\mathcal{F}$. In particular, $H$ is called an $\{P_2,P_3\}$-factor of $G$ if $\mathcal{F}=\{P_2,P_3\}$; $H$ is called an $\mathcal{S}_n$-factor of $G$ if $\mathcal{F}=\{K_{1,1},K_{1,2},K_{1,3},...,K_{1,n}\}$, where $n\geq2$. A spanning subgraph of a graph $G$ is called a $\mathcal{P}_{\geq k}$-factor of $G$ if its each component is isomorphic to a path of order at least $k$, where $k\geq2$. A graph $G$ is called a $\mathcal{F}$-factor covered graph if there is a $\mathcal{F}$-factor of $G$ including $e$ for any $e\in E(G)$. In this paper, we give a minimum degree condition for a $K_{1,r}$-free graph to have an $\mathcal{S}_n$-factor and a $\mathcal{P}_{\geq 3}$-factor, respectively. Further, we obtain sufficient conditions for $K_{1,r}$-free graphs to be $\mathcal{P}_{\geq 2}$-factor, $\mathcal{P}_{\geq 3}$-factor or $\{P_2,P_3\}$-factor covered graphs. In addition, examples show that our results are sharp.

math.CO↗
arXiv2016

Spectrum of Navier $p$-biharmonic problem with sign-changing weight

In this paper, we consider the following eigenvalue problem {{l} (|u"|^{p-2}u")"=λm(x)|u|^{p-2}u, x\in (0,1), u(0)=u(1)=u"(0)=u"(1)=0, where $1<p<+\infty$, $λ$ is a real parameter and $m$ is sign-changing weight. We prove there exists a unique sequence of eigenvalues for above problem. Each eigenvalue is simple and continuous with respect to $p$, the $k$-th eigenfunction, corresponding to the $k$-th positive or negative eigenvalue, has exactly $k-1$ generalized simple zeros in $(0,1)$.

math.CA↗
arXiv2015

Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros

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.

math.AP↗
arXiv2014

Two Whyburn type topological theorems and its applications to Monge-Ampère equations

In this paper we correct a gap of Whyburn type topological lemma and establish two superior limit theorems. As the applications of our Whyburn type topological theorems, we study the following Monge-Ampère equation \begin{eqnarray} \left\{ \begin{array}{lll} \det\left(D^2u\right)=λ^N a(x)f(-u)\,\, &\text{in}\,\, Ω,\\ u=0~~~~~~~~~~~~~~~~~~~~~~\,\,&\text{on}\,\, \partial Ω. \end{array} \right.\nonumber \end{eqnarray} We establish global bifurcation results for the problem. We find intervals of $λ$ for the existence, multiplicity and nonexistence of strictly convex solutions for this problem.

math.FA↗
arXiv2014

Eigenvalue, global bifurcation and positive solutions for a class of fully nonlinear problems

In this paper, we shall study global bifurcation phenomenon for the following Kirchhoff type problem \begin{equation} \left\{ \begin{array}{l} -\left(a+b\int_Ω\vert \nabla u\vert^2\,dx\right)Δu=λu+h(x,u,λ)\,\,\text{in}\,\, Ω,\\ u=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{on}\,\,Ω. \end{array} \right.\nonumber \end{equation} Under some natural hypotheses on $h$, we show that $\left(aλ_1,0\right)$ is a bifurcation point of the above problem. As applications of the above result, we shall determine the interval of $λ$, in which there exist positive solutions for the above problem with $h(x,u;λ)=λf(x,u)-λu$, where $f$ is asymptotically linear at zero and is asymptotically 3-linear at infinity. To study global structure of bifurcation branch, we also establish some properties of the first eigenvalue for a nonlocal eigenvalue problem. Moreover, we also provide a positive answer to an open problem involving the case of $a=0$.

math.AP↗
arXiv2012

Unilateral global bifurcation for fourth-order eigenvalue problems with sign-changing weight

In this paper, we shall establish the unilateral global bifurcation result for a class of fourth-order eigenvalue problems with sign-changing weight. Under some natural hypotheses on perturbation function, we show that $(μ_k^ν,0)$ is a bifurcation point of the above problems and there are two distinct unbounded continua, $(\mathcal{C}_{k}^ν)^+$ and $(\mathcal{C}_{k}^ν)^-$, consisting of the bifurcation branch $\mathcal{C}_{k}^ν$ from $(μ_k^ν, 0)$, where $μ_k^ν$ is the $k$-th positive or negative eigenvalue of the linear problem corresponding to the above problems, $ν\in{+,-}$. As the applications of the above result, we study the existence of nodal solutions for a class of fourth-order eigenvalue problems with sign-changing weight. Moreover, we also establish the Sturm type comparison theorem for fourth-order problems with sign-changing weight.

math.CA↗
arXiv2012

Eigenvalue, bifurcation, existence and nonexistence of solutions for Monge-Ampère equations

In this paper we study the following eigenvalue boundary value problem for Monge-Ampère equations: {equation} \{{array}{l} \det(D^2u)=λ^N f(-u)\,\, \text{in}\,\, Ω, u=0,\,\text{on}\,\, \partial Ω. {array}. {equation} We establish the unilateral global bifurcation results for the problem with $f(u)=u^N+g(u)$ and $Ω$ being the unit ball of $\mathbb{R}^N$. More precisely, under some natural hypotheses on the perturbation function $g:\mathbb{R}\rightarrow\mathbb{R}$, we show that $(λ_1,0)$ is a bifurcation point of the problem and there are two distinct unbounded continua of one-sign solutions, where $λ_1$ is the first eigenvalue of the problem with $f(u)=u^N$. As the applications of the above results, we consider with determining interval of $λ$, in which there exist solutions for this problem in unit ball. Moreover, we also get some results on the existence and nonexistence of convex solutions for this problem in general domain by domain comparison method.

math.AP↗
arXiv2012

Eigenvalues, bifurcation and one-sign solutions for the periodic $p$-Laplacian

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 $λ_0^+$ and $λ_0^-$. Furthermore, under some natural hypotheses on perturbation function, we show that $(λ_0^ν,0)$ is a bifurcation point of the above problems and there are two distinct unbounded sub-continua $\mathscr{C}_ν^{+}$ and $\mathscr{C}_ν^{-}$, consisting of the continuum $\mathscr{C}_ν$ emanating from $(λ_0^ν, 0)$, where $ν\in\{+,-\}$. 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 $λ$ are also studied.

math.CA↗
arXiv2012

Unilateral global bifurcation and nodal solutions for the $p$-Laplacian with sign-changing weight

In this paper, we shall establish a Dancer-type unilateral global bifurcation result for a class of quasilinear elliptic problems with sign-changing weight. Under some natural hypotheses on perturbation function, we show that $(μ_k^ν(p),0)$ is a bifurcation point of the above problems and there are two distinct unbounded continua, $(\mathcal{C}_{k}^ν)^+$ and $(\mathcal{C}_{k}^ν)^-$, consisting of the bifurcation branch $\mathcal{C}_{k}^ν$ from $(μ_k^ν(p), 0)$, where $μ_k^ν(p)$ is the $k$-th positive or negative eigenvalue of the linear problem corresponding to the above problems, $ν\in\{+,-\}$. As the applications of the above unilateral global bifurcation result, we study the existence of nodal solutions for a class of quasilinear elliptic problems with sign-changing weight. Moreover, based on the bifurcation result of Drábek and Huang (1997) [\ref{DH}], we study the existence of one-sign solutions for a class of high dimensional quasilinear elliptic problems with sign-changing weight.

math.AP↗