期刊：
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS,2020年19(5):2575-2616 ISSN：1534-0392
通讯作者：
Yang, Jun
作者机构：
[Wei, Suting] South China Agr Univ, Dept Math, Guangzhou 510642, Peoples R China.;[Yang, Jun; Wei, Suting] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Yang, Jun] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构：
[Yang, Jun] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
摘要：
We consider the nonlinear problem of inhomogeneous Allen-Cahn equation $ \epsilon^2\Delta u+V(y)\,(1-u^2)\,u = 0\quad \mbox{in}\ \Omega, \qquad \frac {\partial u}{\partial \nu} = 0\quad \mbox{on}\ \partial \Omega, $ where $ \Omega $ is a bounded domain in $ \mathbb R^2 $ with smooth boundary, $ \epsilon $ is a small positive parameter, $ \nu $ denotes the unit outward normal of $ \partial \Omega $, $ V $ is a positive smooth function on $ \bar\Omega $. Let $ \Gamma\subset\Omega $ be a smooth curve dividing $ \Omega $ into two disjoint regions and intersecting orthogonally with $ \partial\Omega $ at exactly two points $ P_1 $ and $ P_2 $. Moreover, by considering $ {\mathbb R}^2 $ as a Riemannian manifold with the metric $ g = V(y)\,({\mathrm d}{y}_1^2+{\mathrm d}{y}_2^2) $, we assume that: the curve $ \Gamma $ is a non-degenerate geodesic in the Riemannian manifold $ ({\mathbb R}^2, g) $, the Ricci curvature of the Riemannian manifold $ ({\mathbb R}^2, g) $ along the normal $ \mathbf{n} $ of $ \Gamma $ is positive at $ \Gamma $, the generalized mean curvature of the submanifold $ \partial\Omega $ in $ ({\mathbb R}^2, g) $ vanishes at $ P_1 $ and $ P_2 $. Then for any given integer $ N\geq 2 $, we construct a solution exhibiting $ N $-phase transition layers near $ \Gamma $ (the zero set of the solution has $ N $ components, which are curves connecting $ \partial\Omega $ and directed along the direction of $ \Gamma $) with mutual distance $ O(\epsilon|\log \epsilon|) $, provided that $ \epsilon $ stays away from a discrete set of values to avoid the resonance of the problem. Asymptotic locations of these layers are governed by a Toda system.
期刊：
Journal of Mathematical Analysis and Applications,2020年491(2):124347 ISSN：0022-247X
通讯作者：
Yang, Jun
作者机构：
[Duan, Lipeng] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Yang, Jun] Guangzhou Univ, Sch Math & Informat Sci, Guangzhou 510006, Peoples R China.
通讯机构：
[Yang, Jun] G;Guangzhou Univ, Sch Math & Informat Sci, Guangzhou 510006, Peoples R China.
关键词：
p-Ginzburg-Landau system;Vortices;Existence and Uniqueness;Stability
摘要：
Given p > 2 for the following coupled p-Ginzburg-Landau model in R-2 -Delta(p)u(+) + [A(+)(vertical bar u(+)vertical bar(2) - t(+2)) + A(0)(vertical bar u(-)vertical bar(2) - t(-2))]u(+) = 0, -Delta(p)u(-) + [A(+)(vertical bar u(-)vertical bar(2) - t(-2)) + A(0)(vertical bar u(+)vertical bar(2) - t(+2))]u(-) = 0, with the constraints A(+), A(-) > 0. A(0)(2) < A(+)A(-) and t(+), t(-) > 0, we consider the existence of symmetric vortex solutions u(x) = (U-p(+)(r)e(in+theta), U-p(-)(r)e(in-theta )with given degree (n(+), n(-)) is an element of Z(2), and then prove the uniqueness and regularity results for the vortex profile (U-p(+), U-p(-)) under more constraint of the parameters. Moreover, we also establish the stability result for second variation of the energy around this vortex profile when we consider the perturbations in a space of radial functions. (C) 2020 Elsevier Inc. All rights reserved.
期刊：
Journal of Differential Equations,2020年269(3):1745-1795 ISSN：0022-0396
通讯作者：
Yang, Jun
作者机构：
[Wei, Suting] South China Agr Univ, Dept Math, Guangzhou 510642, Peoples R China.;[Yang, Jun; Wei, Suting] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Yang, Jun] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构：
[Yang, Jun] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
关键词：
Fife-Greenlee problem;Multiple phase transition layers;Resonance;Toda system
摘要：
We consider the Fife-Greenlee problem epsilon(2)Delta u + (u - a(y)) (1-u(2)) = 0 in Omega, partial derivative u/partial derivative v = 0 on partial derivative Omega, where Omega is a bounded domain in R-2 with smooth boundary, epsilon > 0 is a small parameter, nu denotes the unit outward normal of partial derivative Omega. Let Gamma ={y epsilon Omega : a(y) = 0} be a simple smooth curve intersecting orthogonally with partial derivative Omega at exactly two points and dividing Omega into two disjoint nonempty components. We assume that -1 < a(y) < 1 on Omega and del a not equal 0 on Gamma, and also some admissibility conditions hold for a, Gamma and partial derivative Omega. For any fixed integer N = 2m + 1 >= 3, we will show the existence of a clustered solution uewith N-transition layers near Gamma with mutual distance O(epsilon vertical bar log epsilon vertical bar), provided that estays away from a discrete set of values at which resonance occurs. (C) 2020 Elsevier Inc. All rights reserved.
关键词：
Global Science Press;AAMM;Advances in Applied Mathematics and Mechanics;Optimal control problem;elliptic equation;finite element method;ADMM.
摘要：
In this paper, we propose an efficient numerical method for the optimal control problem constrained by elliptic equations. Being approximated by the finite element method (FEM), the continuous optimal control problem is discretized into a finite dimensional optimization problem with separable structures. Furthermore, an alternating direction method of multipliers (ADMM) is applied to solve the discretization problem. The total convergence analysis which includes the discretization error by FEM and iterative error by ADMM is established. Finally, numerical simulations are presented to verify the efficiency of the proposed method.
期刊：
DIFFERENTIAL AND INTEGRAL EQUATIONS,2019年32(1-2):49-90 ISSN：0893-4983
通讯作者：
Gheraibia, Billel
作者机构：
[Wang, Chunhua; Gheraibia, Billel] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Wang, Chunhua; Gheraibia, Billel] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.;[Yang, Jing] Jiangsu Univ Sci & Technol, Coll Math & Phys, Zhenjiang 212003, Jiangsu, Peoples R China.
通讯机构：
[Gheraibia, Billel] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.
关键词：
In this paper;we study the following Grushin critical problem $$ -\Delta u(x)=\Phi(x)\frac{u^{\frac{N}{N-2}}(x)} {|y|};u>0;\text{in}\;\mathbb R^{N};$$ where $x=(y;z)\in\mathbb R^{k}\times \mathbb R^{N-k};N\geq 5;\Phi(x)$ is positive and periodic in its the $\bar{k}$ variables $(z_{1};z_{\bar{k}});1\leq \bar{k} < \frac{N-2}{2}.$ Under some suitable conditions on $\Phi(x)$ near its critical point;we prove that the problem above has solutions with infinitely many bubbles. Moreover;we also show that the bubbling solutions obtained in our existence result are locally unique. Our result implies that some bubbling solutions preserve the symmetry from the potential $\Phi(x).$ Published: January/February 2019 First available in Project Euclid: 11 December 2018 zbMATH: 07031709 MathSciNet: MR3909979 Digital Object Identifier: 10.57262/die/1544497286 Subjects: Primary: 35B40;35B45;35J40
摘要：
In this paper, we study the following Grushin critical problem $$ -\Delta u(x)=\Phi(x)\frac{u^{\frac{N}{N-2}}(x)} {|y|},\,\,\,\,u>0,\,\,\,\text{in}\,\,\,\mathbb R^{N}, $$ where $x=(y,z)\in\mathbb R^{k}\times \mathbb R^{N-k},N\geq 5,\Phi(x)$ is positive and periodic in its the $\bar{k}$ variables $(z_{1},...,z_{\bar{k}}),1\leq \bar{k} < \frac{N-2}{2}.$ Under some suitable conditions on $\Phi(x)$ near its critical point, we prove that the problem above has solutions with infinitely many bubbles. Moreover, we also show that the bubbling solutions obtained in our existence result are locally unique. Our result implies that some bubbling solutions preserve the symmetry from the potential $\Phi(x).$
期刊：
Journal of Differential Equations,2019年266(9):5821-5866 ISSN：0022-0396
通讯作者：
Yang, Jun
作者机构：
[Fan, Xu-Qian] Jinan Univ, Dept Math, Guangzhou 510632, Guangdong, Peoples R China.;[Xu, Bin] Jiangsu Normal Univ, Sch Math & Stat, Xuzhou 221116, Jiangsu, Peoples R China.;[Yang, Jun] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Yang, Jun] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Yang, Jun] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.
摘要：
We consider the nonlinear problem of inhomogeneous Allen-Cahn equation epsilon(2)Delta u+V(y)u(1-u(2)) = 0 in Omega, partial derivative u/partial derivative v = 0 on partial derivative Omega, where Q is a bounded domain in R-2 with smooth boundary, E is a small positive parameter, v denotes the unit outward normal of partial derivative Omega, V is a positive smooth function on (Omega) over bar. Let Gamma be a curve intersecting orthogonally with partial derivative Omega at exactly two points and dividing Omega into two parts. Moreover, Gamma satisfies stationary and non-degenerate conditions with respect to the functional integral(Gamma) V-1/2. We can prove that there exists a solution u, such that: as epsilon -> 0, u(epsilon) approaches +1 in one part of Omega, while tends to -1 in the other part, except a small neighborhood of Gamma. (C) 2018 Elsevier Inc. All rights reserved.
期刊：
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS,2019年39(4):1745-1777 ISSN：1078-0947
通讯作者：
Yang, Jun
作者机构：
[Jiang, Ruiqi] Hunan Univ, Coll Math & Econometr, Changsha 410082, Hunan, Peoples R China.;[Wang, Youde] Guangzhou Univ, Coll Math & Informat Sci, Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China.;[Yang, Jun] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Yang, Jun] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Yang, Jun] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.
期刊：
Journal of Differential Equations,2019年267(7):4117-4147 ISSN：0022-0396
通讯作者：
Yan, Shusen
作者机构：
[Long, Wei; Yang, Jianfu] Jiangxi Normal Univ, Coll Math & Informat Sci, Nanchang 330022, Jiangxi, Peoples R China.;[Yan, Shusen] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Yan, Shusen] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
关键词：
Fractional Laplacian;Semilinear nonlocal equation;Critical;Blow up
摘要：
Let Omega(epsilon) be a bounded domain in R-N with small holes. In this paper, we study the existence of solutions for the following problem { (-Delta)(s) u = u(2)*(s-1), u > 0, in Omega(epsilon), u = 0, in R-N \ Omega(epsilon,) where 0 < s < 1 and 2(s)*= 2N/N-2s. We construct solutions which blow up like a volcano near the center of each hole in Omega(epsilon) (C) 2019 Elsevier Inc. All rights reserved.
期刊：
Calculus of Variations and Partial Differential Equations,2018年57(3):1-45 ISSN：0944-2669
通讯作者：
Yang, Jun
作者机构：
[Yang, Jun; Wei, Suting] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Xu, Bin] Jiangsu Normal Univ, Sch Math & Stat, Xuzhou 221116, Jiangsu, Peoples R China.;[Yang, Jun] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Yang, Jun] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.
关键词：
35J25;35J61
摘要：
We consider the problem
$$\begin{aligned} \epsilon ^2 \Delta u-V(y)u+u^p\,=\,0,\quad u>0\quad \text{ in }\quad \Omega , \quad \frac{\partial u}{\partial \nu }\,=\,0\quad \text{ on }\quad \partial \Omega , \end{aligned}$$
where
$$\Omega $$
is a bounded domain in
$${\mathbb {R}}^2$$
with smooth boundary, the exponent p is greater than 1,
$$\epsilon >0$$
is a small parameter, V is a uniformly positive, smooth potential on
$$\bar{\Omega }$$
, and
$$\nu $$
denotes the outward unit normal of
$$\partial \Omega $$
. Let
$$\Gamma $$
be a curve intersecting orthogonally
$$\partial \Omega $$
at exactly two points and dividing
$$\Omega $$
into two parts. Moreover,
$$\Gamma $$
satisfies stationary and non-degeneracy conditions with respect to the functional
$$\int _{\Gamma }V^{\sigma }$$
, where
$$\sigma =\frac{p+1}{p-1}-\frac{1}{2}$$
. We prove the existence of a solution
$$u_\epsilon $$
concentrating along the whole of
$$\Gamma $$
, exponentially small in
$$\epsilon $$
at any fixed distance from it, provided that
$$\epsilon $$
is small and away from certain critical numbers. In particular, this establishes the validity of the two dimensional case of a conjecture by Ambrosetti et al. (Indiana Univ Math J 53(2), 297–329, 2004).
期刊：
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS,2018年38(3):1527-1552 ISSN：1078-0947
通讯作者：
Yang, Jun
作者机构：
[Tang, Feifei; Yang, Jun; Wei, Suting] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Yang, Jun] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Yang, Jun] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.
摘要：
We consider the Fife-Greenlee problem $ε^2\triangle u + \bigl(u-\mathbf{a}(y)\bigr)(1-u^2) =0 ~~~ \mbox{in}\ Ω,~~~~~~~\frac{\partial u}{\partialν} = 0 ~~~ \mbox{on}\ \partialΩ,$ where $Ω$ is a bounded domain in ${\mathbb R}^2$ with smooth boundary, $\epsilon>0$ is a small parameter, $ν$ denotes the unit outward normal of $\partialΩ$. Let $Γ = \{y∈ Ω: \mathbf{a}(y) = 0 \}$ be a simple smooth curve intersecting orthogonally with $\partialΩ$ at exactly two points and dividing $Ω$ into two disjoint nonempty components. We assume that $-1\,<\,\mathbf{a}(y)\,<1$ on $Ω$ and $\triangledown\mathbf{a}≠ 0$ on $Γ$, and also some admissibility conditions between the curves $Γ$, $\partialΩ$ and the inhomogeneity ${\mathbf a}$ hold at the connecting points. We can prove that there exists a solution $u_{\epsilon}$ such that: as $\epsilon → 0$, $u_{\epsilon}$ approaches $+1$ in one part, while tends to $-1$ in the other part, except a small neighborhood of $Γ$.
期刊：
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS,2018年38(11):5461-5504 ISSN：1078-0947
通讯作者：
Wang, Chunhua
作者机构：
[Wang, Chunhua] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Wang, Chunhua] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.;[Yang, Jing] Jiangsu Univ Sci & Technol, Sch Sci, Zhenjiang 212003, Peoples R China.
通讯机构：
[Wang, Chunhua] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.
关键词：
Nonsymmetric potential;Reduction;Schrödinger-Poisson system
摘要：
<p style='text-indent:20px;'>In this paper, we study the following nonlinear Schrödinger-Poisson system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$\left\{\begin{array}{ll} -\Delta u+u+\epsilon K(x)\Phi(x)u = f(u),& x\in \mathbb{R}^{3} , \\ -\Delta \Phi = K(x)u^{2},\,\,& x\in \mathbb{R}^{3}, \\\end{array}\right.$ \end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1"> \begin{document} $K(x)$ \end{document}</tex-math></inline-formula> is a positive and continuous potential and <inline-formula><tex-math id="M2"> \begin{document} $f(u)$ \end{document}</tex-math></inline-formula> is a nonlinearity satisfying some decay condition and some non-degeneracy condition, respectively. Under some suitable conditions, which are given in section 1, we prove that there exists some <inline-formula><tex-math id="M3"> \begin{document} $\epsilon_{0}>0$ \end{document}</tex-math></inline-formula> such that for <inline-formula><tex-math id="M4"> \begin{document} $0<\epsilon<\epsilon_{0}$ \end{document}</tex-math></inline-formula>, the above problem has infinitely many positive solutions by applying localized energy method. Our main result can be viewed as an extension to a recent result Theorem 1.1 of Ao and Wei in [<xref ref-type="bibr" rid="aw">3</xref>] and a result of Li, Peng and Wang in [<xref ref-type="bibr" rid="lpw">26</xref>].</p>
期刊：
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS,2016年36(3):1603-1628 ISSN：1078-0947
通讯作者：
Wang, Chunhua
作者机构：
[Wang, Chunhua; Yang, Jing] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Wang, Chunhua] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Double critical Hardy-Sobolev-Maz'ya terms;Infinitely many solutions;Variational methods
摘要：
In this paper, we investigate the following elliptic problem involving double critical Hardy-Sobolev-Maz'ya terms: $$ \left\{\begin{array}{ll} -\Delta u = \mu\frac{|u|^{2^*(t)-2}u}{|y|^t} + \frac{|u|^{2^*(s)-2}u}{|y|^s} + a(x) u, & {\rm in}\ \Omega,\\ \quad u = 0, \,\, &{\rm on}\ \partial \Omega, \end{array} \right. $$ where $\mu\geq0$, $a(x)>0$, $2^*(t)=\frac{2(N-t)}{N-2}$, $2^*(s) = \frac{2(N-s)}{N-2}$, $0\leq t<s<2$, $x = (y,z)\in \mathbb{R}^k\times \mathbb{R}^{N-k}$, $2\leq k<N$, $(0,z^*) \in \bar{\Omega}$ and $\Omega$ is an bounded domain in $\mathbb{R}^N$. Applying an abstract theorem in \cite{sz}, we prove that if $N>6+t$ when $\mu>0,$ and $N>6+s$ when $\mu=0,$ and $\Omega$ satisfies some geometric conditions, then the above problem has infinitely many sign-changing solutions. The main tool is to estimate Morse indices of these nodal solution.
作者机构：
[Peng, Shuangjie; Long, Wei; Yang, Jing] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Long, Wei] Jiangxi Normal Univ, Coll Math & Informat Sci, Nanchang 330022, Jiangxi, Peoples R China.
通讯机构：
[Long, Wei] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
会议地点：
Xiamen Univ, Xiamen, PEOPLES R CHINA
会议主办单位：
Xiamen Univ
关键词：
Fractional Laplacian;Nonlinear scalar field equation;Reduction method
摘要：
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.
期刊：
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY,2016年368(4):2589-2622 ISSN：0002-9947
通讯作者：
Wei, Juncheng;Yang, Jun
作者机构：
[Wei, Juncheng] Univ British Columbia, Dept Math, Vancouver, BC V6T 1Z2, Canada.;[Wei, Juncheng] Chinese Univ Hong Kong, Dept Math, Shatin, Hong Kong, Peoples R China.;[Yang, Jun] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Yang, Jun] Shenzhen Univ, Coll Math & Computat Sci, Nanhai Ave 3688, Shenzhen 518060, Peoples R China.
通讯机构：
[Wei, Juncheng] U;[Wei, Juncheng; Yang, Jun] C;[Yang, Jun] S;Univ British Columbia, Dept Math, Vancouver, BC V6T 1Z2, Canada.;Chinese Univ Hong Kong, Dept Math, Shatin, Hong Kong, Peoples R China.
摘要：
We construct traveling wave solutions with vortex helix structures for the Schrodinger map equation partial derivative m/partial derivative t = m x (Delta m - m3 (e) over right arrow3) on R-3 x R of the form m(s(1), s(2), s(3) - delta vertical bar log epsilon vertical bar epsilon t) with traveling velocity delta vertical bar log epsilon vertical bar epsilon along the direction of the s3 axis. We use a perturbation approach which gives a complete characterization of the asymptotic behavior of the solutions.
作者机构：
[Long, Wei] Jiangxi Normal Univ, Coll Math & Informat Sci, Nanchang 330022, Jiangxi, Peoples R China.;[Wang, Qingfang; Yang, Jing] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Long, Wei] J;Jiangxi Normal Univ, Coll Math & Informat Sci, Nanchang 330022, Jiangxi, Peoples R China.
摘要：
This paper is concerned with the following nonlinear fractional Schrödinger equation where is a small parameter, is a positive function, and . Under some suitable conditions, we prove that for any positive integer , one can construct a -spike positive solution near the local maximum point of .
期刊：
ADVANCES IN DIFFERENTIAL EQUATIONS,2015年20(1-2):77-116 ISSN：1079-9389
通讯作者：
Wang, Chunhua
作者机构：
[Wang, Chunhua; Wang, Qingfang; Yang, Jing] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Wang, Chunhua] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词：
By a Lyapunov-Schmidt reduction argument;for a type of Grushin critical problem with cylindrical symmetry;we prove that it has infinitely many positive solutions with cylindrical symmetry;whose energy can be made arbitrarily large. Published: January/February 2015 First available in Project Euclid: 11 December 2014 zbMATH: 1311.35169 MathSciNet: MR3297780 Digital Object Identifier: 10.57262/ade/1418310443 Subjects: Primary: 35B40;35B45;35J40
摘要：
By a Lyapunov-Schmidt reduction argument, for a type of Grushin critical problem with cylindrical symmetry, we prove that it has infinitely many positive solutions with cylindrical symmetry, whose energy can be made arbitrarily large.
期刊：
Journal of Mathematical Physics,2015年56(5):051505 ISSN：0022-2488
通讯作者：
Wang, Chunhua
作者机构：
[Wang, Chunhua; Yang, Jing] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Wang, Chunhua] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Schrodinger equation
摘要：
We study a linearly coupled Schr"dinger system in R-N (N <= 3). Assume that the potentials in the system are continuous functions satisfying suitable decay assumptions, but without any symmetry properties, and the parameters in the system satisfy some suitable restrictions. Using the Liapunov-Schmidt reduction methods two times and combining localized energy method, we prove that the problem has infinitely many positive synchronized solutions, which extends result Theorem 1.2 about nonlinearly coupled Schrdinger equations in Ao and Wei [Calculus Var. Partial Differ. Equations 51, 761-798 (2014)] to our linearly coupled problem. (C) 2015 AIP Publishing LLC.
摘要：
Let Omega be a bounded domain with a smooth C-2 boundary in R-N = R-k x RN-k (N >= 3), 0 is an element of partial derivative Omega, and v denotes the unit outward normal vector to boundary as partial derivative Omega. We are concerned with the Neumann boundary problem: -Delta u-mu(vertical bar y vertical bar 2)/(u) = (vertical bar y vertical bar t)/(vertical bar u vertical bar Pt-1u) + f(x,u), u > 0, x is an element of Omega, partial derivative v/partial derivative u + alpha(x)u = 0, x is an element of partial derivative Omega \ {0}. Using the Mountain Pass Lemma without (PS) condition and the strong maximum principle, we establish certain existence result of the positive solutions. (C) 2014 Elsevier Inc. All rights reserved.
作者机构：
[Yang, Jing; Peng, Shuang Jie] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Peng, Shuang Jie] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Hardy-Sobolev-Maz’ya inequality;Mountain Pass Lemma;positive solutions;subsolution and supersolution
摘要：
In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy-Sobolev-Maz’ya term:
$ - \Delta u - \lambda \frac{u}
{{|y|^2 }} = \frac{{|u|^{p_t - 1} u}}
{{|y|^t }} + \mu f(x),x \in \Omega ,$
where Ω is a bounded domain in ℝ
N
(N ≥ 2), 0 ∈ Ω, x = (y, z) ∈ ℝ
k
× ℝ
N-k
and
$p_t = \frac{{N + 2 - 2t}}
{{N - 2}}(0 \leqslant t \leqslant 2)$
For f(x) ∈ C
1(
$\bar \Omega $
){0}, we show that there exists a constant μ* > 0 such that the problem possesses at least two positive solutions if μ ∈ (0, μ*) and at least one positive solution if μ = μ*. Furthermore, there are no positive solutions if μ ∈ (μ*,+∞).
