CPAA
Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents
Yanfang Peng Jing Yang
Communications on Pure & Applied Analysis 2015, 14(2): 439-455 doi: 10.3934/cpaa.2015.14.439
We study the following elliptic equation with two Sobolev-Hardy critical exponents \begin{eqnarray} & -\Delta u=\mu\frac{|u|^{2^{*}(s_1)-2}u}{|x|^{s_1}}+\frac{|u|^{2^{*}(s_2)-2}u}{|x|^{s_2}} \quad x\in \Omega, \\ & u=0, \quad x\in \partial\Omega, \end{eqnarray} where $\Omega\subset R^N (N\geq3)$ is a bounded smooth domain, $0\in\partial\Omega$, $0\leq s_2 < s_1 \leq 2$ and $2^*(s):=\frac{2(N-s)}{N-2}$. In this paper, by means of variational methods, we obtain the existence of sign-changing solutions if $H(0)<0$, where $H(0)$ denote the mean curvature of $\partial\Omega$ at $0$.
keywords: mean curvature sign-changing solutions. Critical exponents
CPAA
Segregated vector Solutions for nonlinear Schrödinger systems with electromagnetic potentials
Jing Yang
Communications on Pure & Applied Analysis 2017, 16(5): 1785-1805 doi: 10.3934/cpaa.2017087
In this paper, we study the following nonlinear Schrödinger system in
$\mathbb{R}^3$
$\left\{ \begin{array}{*{35}{l}} {{(\frac{\nabla }{i}-A(y))}^{2}}u+{{\lambda }_{1}}(|y|)u=|u{{|}^{2}}u+\beta |v{{|}^{2}}u,&x\in {{\mathbb{R}}^{3}}, \\ {{(\frac{\nabla }{i}-A(y))}^{2}}v+{{\lambda }_{2}}(|y|)v=|v{{|}^{2}}v+\beta |u{{|}^{2}}v,&x\in {{\mathbb{R}}^{3}}, \\\end{array} \right.$
where
$A(y)=A(|y|)∈ C^1(\mathbb{R}^3,\mathbb{R})$
is bounded,
$λ_1(|y|),λ_2(|y|)$
are continuous positive radial potentials, and
$β∈ \mathbb{R}$
is a coupling constant. We proved that if
$A(y),λ_1(y),λ_2(y)$
satisfy some suitable conditions, the above problem has infinitely many non-radial segregated solutions.
keywords: Segregated vector solutions electromagnetic fields reduction method
DCDS
Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations
Wei Long Shuangjie Peng Jing Yang
Discrete & Continuous Dynamical Systems - A 2016, 36(2): 917-939 doi: 10.3934/dcds.2016.36.917
We consider the following nonlinear fractional scalar field equation $$ (-\Delta)^s u + u = K(|x|)u^p,\ \ u > 0 \ \ \hbox{in}\ \ \mathbb{R}^N, $$ where $K(|x|)$ is a positive radial function, $N\ge 2$, $0 < s < 1$, and $1 < p < \frac{N+2s}{N-2s}$. Under various asymptotic assumptions on $K(x)$ at infinity, we show that this problem has infinitely many non-radial positive solutions and sign-changing solutions, whose energy can be made arbitrarily large.
keywords: Fractional Laplacian reduction method. nonlinear scalar field equation
KRM
Global existence and steady states of a two competing species Keller--Segel chemotaxis model
Qi Wang Lu Zhang Jingyue Yang Jia Hu
Kinetic & Related Models 2015, 8(4): 777-807 doi: 10.3934/krm.2015.8.777
We study a one--dimensional quasilinear system proposed by J. Tello and M. Winkler [27] which models the population dynamics of two competing species attracted by the same chemical. The kinetic terms of the interacting species are chosen to be of the Lotka--Volterra type and the boundary conditions are of homogeneous Neumann type which represent an enclosed domain. We prove the global existence and boundedness of classical solutions to the fully parabolic system. Then we establish the existence of nonconstant positive steady states through bifurcation theory. The stability or instability of the bifurcating solutions is investigated rigorously. Our results indicate that small intervals support stable monotone positive steady states and large intervals support nonmonotone steady states. Finally, we perform extensive numerical studies to demonstrate and verify our theoretical results. Our numerical simulations also illustrate the formation of stable steady states and time--periodic solutions with various interesting spatial structures.
keywords: two species chemotaxis model. Global existence stationary solutions
DCDS
Infinitely many solutions for an elliptic problem with double critical Hardy-Sobolev-Maz'ya terms
Chunhua Wang Jing Yang
Discrete & Continuous Dynamical Systems - A 2016, 36(3): 1603-1628 doi: 10.3934/dcds.2016.36.1603
In this paper, by an approximating argument, we obtain infinitely many solutions for the following problem \begin{equation*} \left\{ \begin{array}{ll} -\Delta u = \mu \frac{|u|^{2^{*}(t)-2}u}{|y|^{t}} + \frac{|u|^{2^{*}(s)-2}u}{|y|^{s}} + a(x) u, & \hbox{$\text{in} \Omega$}, \\ u=0,\,\, &\hbox{$\text{on}~\partial \Omega$}, \\ \end{array} \right. \end{equation*} where $\mu\geq0,2^{*}(t)=\frac{2(N-t)}{N-2},2^{*}(s) = \frac{2(N-s)}{N-2},0\leq t < s < 2,x = (y,z)\in \mathbb{R}^{k} \times \mathbb{R}^{N-k},2 \leq k < N,(0,z^*)\subset \bar{\Omega}$ and $\Omega$ is an open bounded domain in $\mathbb{R}^{N}.$ We prove that if $N > 6+t$ when $\mu>0$ and $N > 6+s$ when $\mu=0,$ $a((0,z^*)) > 0,$ $\Omega$ satisfies some geometric conditions, then the above problem has infinitely many solutions.
keywords: variational methods. infinitely many solutions Double critical Hardy-Sobolev-Maz'ya terms
DCDS-B
Time-periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: Effect of cellular growth
Qi Wang Jingyue Yang Lu Zhang
Discrete & Continuous Dynamical Systems - B 2017, 22(9): 3547-3574 doi: 10.3934/dcdsb.2017179

This paper investigates the formation of time-periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model with a focus on the effect of cellular growth. We carry out rigorous Hopf bifurcation analysis to obtain the bifurcation values, spatial profiles and time period associated with these oscillating patterns. Moreover, the stability of the periodic solutions is investigated and it provides a selection mechanism of stable time-periodic mode which suggests that only large domains support the formation of these periodic patterns. Another main result of this paper reveals that cellular growth is responsible for the emergence and stabilization of the oscillating patterns observed in the 3×3 system, while the system admits a Lyapunov functional in the absence of cellular growth. Global existence and boundedness of the system in 2D are proved thanks to this Lyapunov functional. Finally, we provide some numerical simulations to illustrate and support our theoretical findings.

keywords: Two species chemotaxis model time-periodic pattern stable pattern Hopf bifurcation stability analysis Lyapunov functional
CPAA
Liouville type theorem to an integral system in the half-space
Weiwei Zhao Jinge Yang Sining Zheng
Communications on Pure & Applied Analysis 2014, 13(2): 511-525 doi: 10.3934/cpaa.2014.13.511
In this paper, by using the moving plane method in integral forms, we establish a Liouville type theorem for a coupled integral system with Navier boundary values in the half-space. Furthermore, we prove that the Liouville type theorem is valid for the related differential system as well under an additional assumption by showing the equivalence between the involved differential and integral systems.
keywords: Navier boundary value Liouville type theorem Kelvin transformation. Rotational symmetry Moving plane method in integral forms
DCDS
Boundedness in logistic Keller-Segel models with nonlinear diffusion and sensitivity functions
Qi Wang Jingyue Yang Feng Yu
Discrete & Continuous Dynamical Systems - A 2017, 37(9): 5021-5036 doi: 10.3934/dcds.2017216
We consider the following fully parabolic Keller-Segel system
$\left\{\begin{array}{ll}u_t=\nabla · (D(u) \nabla u-S(u) \nabla v)+u(1-u^γ),&x ∈ Ω,t>0, \\v_t=Δ v-v+u,&x ∈ Ω,t>0, \\\frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}=0,&x∈\partial Ω,t>0\end{array}\right.$
over a multi-dimensional bounded domain
$Ω \subset \mathbb R^N$
,
$N≥2$
. Here
$D(u)$
and
$S(u)$
are smooth functions satisfying:
$D(0)>0$
,
$D(u)≥ K_1u^{m_1}$
and
$S(u)≤ K_2u^{m_2}$
,
$\forall u≥0$
, for some constants
$K_i∈\mathbb R^+$
,
$m_i∈\mathbb R$
,
$i=1, 2$
. It is proved that, when the parameter pair
$(m_1, m_2)$
lies in some specific regions, the system admits global classical solutions and they are uniformly bounded in time. We cover and extend [22,28], in particular when
$N≥3$
and
$γ≥1$
, and [3,29] when
$m_1>γ-\frac{2}{N}$
if
$γ∈(0, 1)$
or
$m_1>γ-\frac{4}{N+2}$
if
$γ∈[1, ∞)$
. Moreover, according to our results, the index
$\frac{2}{N}$
is, in contrast to the model without cellular growth, no longer critical to the global existence or collapse of this system.
keywords: Chemotaxis nonlinear diffusion global existence boundedness logistic growth

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