作者机构：
[Peng, Shuangjie; Long, Wei] 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：
[Peng, Shuangjie] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
摘要：
This paper is concerned with the existence of segregated vector solutions for [GRAPHICS] where Omega is a bounded or unbounded domain in R-N with N = 1, 2, 3, epsilon > 0 is a small parameter, n > 1 is an integer, P-i (i=1,...,n) is a potential function, beta(i) > 0 (i = 1,..., n) is constant and beta(ij) =beta(ji) > 0 (j not equal i) is coupling constant. This system describes some physical phenomena such as the propagation in birefringent optical fibers, Kerr-like photorefractive in optics and Bose-Einstein condensates. For beta(ij) > 0, which corresponds to the synchronization case for the above system with constant potentials, we prove that the system has multiple positive vector solutions, whose components may have spikes clustering at the same point as epsilon -> 0(+), but the distance between them divided by a will go to infinity. (C) 2014 Elsevier Inc. All rights reserved.
摘要：
In this paper, by an approximating argument, we obtain infinitely many radial solutions for the following elliptic systems with critical Sobolev growth { -Delta u - vertical bar u vertical bar(2)*(-2) u + eta alpha/alpha+beta vertical bar u vertical bar alpha-2 u vertical bar v vertical bar(beta) + sigma p/p+q vertical bar u vertical bar(p-2) u vertical bar v vertical bar(q,) x is an element of B, -Delta v = vertical bar v vertical bar(2)*(-2) v + eta beta/alpha+beta vertical bar v vertical bar alpha-2 u vertical bar v vertical bar(beta) + sigma q/p+q vertical bar u vertical bar(p) vertical bar v vertical bar(q-2) x is an element of B, u = v = 0, where N > 2 (p+q+1)/p+q-1, eta, sigma > 0, alpha, beta > 1 and alpha + beta = 2* =: 2N/N-2, p,q >= 1, 2 <= p + q < 2* and B C R-N is an open ball centered at the origin.
摘要：
We consider the system linearly coupled by nonlinear Schrödinger equations in ℝ3: $\{\begin{array}{c}-\mathrm{\Delta }{\mathrm{u}}_{\mathrm{j}}+{\mathrm{u}}_{\mathrm{j}}={\mathrm{u}}_{\mathrm{j}}^{3}-\mathrm{\varepsilon }\sum _{\mathrm{i}\ne \mathrm{j}}^{\mathrm{N}}{\mathrm{u}}_{\mathrm{i}},\mathrm{x}\in {\mathrm{\mathbb{R}}}^{3},\\ {\mathrm{u}}_{\mathrm{j}}\in {\mathrm{H}}^{1}\left({\mathrm{\mathbb{R}}}^{3}\right),\mathrm{j}=1,\cdots,\mathrm{N},\end{array}$ where ε ∈ ℝ is a coupling constant. This type of system arises in particular in models in nonlinear N-core fiber. We then examine how the linear coupling affects the solution structure. When N = 2,3, for any prescribed integer ℓ ≥ 2, we construct a nonradial vector solution of segregated type, with two components having exactly ℓ positive bumps for ε > 0 sufficiently small. We also give an explicit description of the characteristic features of the vector solutions.
作者机构：
[Peng, Shuangjie; Deng, Yinbin] Huazhong Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Wang, Jixiu] Hubei Univ Arts & Sci, Sch Math & Comp Sci, Xiangyang 441053, Peoples R China.
通讯机构：
[Deng, Yinbin] H;Huazhong Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词：
minimisation, Schrodinger equation, solitons
摘要：
This paper is concerned with constructing nodal radial solutions for generalized quasilinear Schrodinger equations in R-N which arise from plasma physics, fluid mechanics, as well as high-power ultashort laser in matter. For any given integer k >= 0, by using a change of variables and minimization argument, we obtain a sign-changing minimizer with k nodes of a minimization problem. (c) 2014 AIP Publishing LLC.
摘要：
We consider the existence of positive bound states for the nonlinear Schrodinger equation -epsilon(2)Delta u + V(x)u = u(p), u > 0, u is an element of H-1(R-N), where N >= 3, 1 < p < (N + 2)/(N - 2), and V is a nonnegative potential with compact support. For arbitrary positive integer k is an element of Z(+), we construct higher energy solutions with exactly k peaks which interact with each other and cluster around a local maximum point of V when e is sufficiently small. The main part of the solutions decays exponentially but the perturbation part decays algebraically at infinity.
摘要：
We study the following singular elliptic equation -div(vertical bar x vertical bar(-2a) del u) - mu u/vertical bar x vertical bar(2(1+a)) = vertical bar u vertical bar(p-2)u/vertical bar x vertical bar(bp) + lambda u/vertical bar x vertical bar(dD) with Dirichlet boundary condition, which is related to the well-known Caffarelli-Kohn-Nirenberg inequalities. By virtue of variational method and Nehari manifold, we obtain least energy sign-changing solutions in some ranges of the parameters mu and lambda. In particular, our result generalizes the existence results of sign-changing solutions to lower dimensions 5 and 6.
作者机构：
[Li, Gongbao] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Gongbao] C;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
摘要：
We study the concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth, -(epsilon(2)a + epsilon b integral(R3) vertical bar del u vertical bar(2))del u + V(z)u = f(u) + u(5) in R-3, u is an element of H-1(R-3), u > 0 in R-3, where e is a small positive parameter and a, b > 0 are constants, f is an element of C-1(R+, R) is subcritical, V : R-3 -> R is a locally Holder continuous function. We first prove that for epsilon(0) > 0 sufficiently small, the above problem has a weak solution u(epsilon) with exponential decay at infinity. Moreover, u(epsilon) concentrates around a local minimum point of V in Lambda as epsilon -> 0. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple solutions by employing the topological construction of the set where the potential V(z) attains its minimum.
摘要：
This paper is concerned with constructing nodal radial solutions for quasilinear Schrodinger equations in R-N with critical growth which have appeared as several models in mathematical physics. For any given integer k >= 0, by using a change of variables and minimization argument, we obtain a sign-changing minimizer with k nodes of a minimization problem. Since the critical exponent appears and the lower order term may change sign, we should use more delicate arguments. (C) 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4774153]
作者机构：
[Cao, Daomin] Chinese Acad Sci, Acad Math & Syst Sci, Inst Appl Math, Beijing 100190, Peoples R China.;[Peng, Shuangjie] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Cao, Daomin] C;Chinese Acad Sci, Acad Math & Syst Sci, Inst Appl Math, Beijing 100190, Peoples R China.
摘要：
By variational methods, for a kind of Webster scalar curvature problems on the CR sphere with cylindrically symmetric curvature, we construct some multi-peak solutions as the parameter is sufficiently small under certain assumptions. We also obtain the asymptotic behaviors of the solutions.
作者机构：
[Peng, Shuangjie; Wang, Zhi-qiang] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;Nankai Univ, Chern Inst Math, Tianjin 300071, Peoples R China.;[Wang, Zhi-qiang] Utah State Univ, Dept Math & Stat, Logan, UT 84322 USA.
通讯机构：
[Wang, Zhi-qiang] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
摘要：
We consider the following nonlinear Schrödinger system in
$${\mathbb{R}^3}$$
$$\left\{\begin{array}{ll}-\Delta u + P(|x|)u = \mu u^{2}u + \beta v^2u,\quad x \in \mathbb{R}^3,\\-\Delta v + Q(|x|)v = \nu v^{2}v + \beta u^2v,\quad x \in \mathbb{R}^3,\end{array}\right.$$
where P(r) and Q(r) are positive radial potentials,
$${\mu > 0, \nu > 0}$$
and
$${\beta \in \mathbb{R}}$$
is a coupling constant. This type of system arises, in particular, in models in Bose–Einstein condensates theory. We examine the effect of nonlinear coupling on the solution structure. In the repulsive case, we construct an unbounded sequence of non-radial positive vector solutions of segregated type, and in the attractive case we construct an unbounded sequence of non-radial positive vector solutions of synchronized type. Depending upon the system being repulsive or attractive, our results exhibit distinct characteristic features of vector solutions.
作者机构：
[Kang DongSheng] S Cent Univ Nationalities, Sch Math & Stat, Wuhan 430074, Peoples R China.;[Peng ShuangJie] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Kang DongSheng] S;S Cent Univ Nationalities, Sch Math & Stat, Wuhan 430074, Peoples R China.
摘要：
In this paper, a system of elliptic equations is investigated, which involves Hardy potential and multiple critical Sobolev exponents. By a global compactness argument of variational method and a fine analysis on the Palais-Smale sequences created from related approximation problems, the existence of infinitely many solutions to the system is established.
作者机构：
[Cao, Daomin] Chinese Acad Sci, Inst Appl Math, AMSS, Beijing 100190, Peoples R China.;[Peng, Shuangjie] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Cao, Daomin] Chinese Acad Sci, Key Lab Random Complex Struct & Data Sci, Beijing 100190, Peoples R China.;[Yan, Shusen] Univ New England, Dept Math, Armidale, NSW 2351, Australia.
通讯机构：
[Peng, Shuangjie] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Critical Sobolev growth;Infinitely many solutions;p-Laplacian equations
摘要：
In this paper, we will prove the existence of infinitely many solutions for the following elliptic problem with critical Sobolev growth: -Delta(p)u = vertical bar u vertical bar(p)*(-2)u + mu vertical bar u vertical bar(p-2)u in Omega, u =0 on partial derivative Omega, provided N > p(2) + p, where Delta(p) is the p-Laplacian operator, 1 < p < N, p* = pN/N-p, mu > 0 and Omega is an open bounded domain in R-N. (C) 2012 Elsevier Inc. All rights reserved.
摘要：
In this paper, we are concerned with the following nonlinear Schrodinger equations with inverse square potential and critical Sobolev exponent -Delta u - mu u/vertical bar x vertical bar(2) + a(x)u = vertical bar u vertical bar(2)*(-2)u + f(x, u), u is an element of H-1(R-N), (P) where 2* = 2N/(N - 2) is the critical Sobolev exponent, 0 <= mu < <(mu)over bar> := (N-2)(2)/4, a(x) is an element of C(R-N). We first give a representation to the Palais-Smale sequence related to (P) and then obtain an existence result of positive solutions of (P). Our assumptions on a(x) and f(x, u) are weaker than the known cases even if mu = 0. (C) 2012 Elsevier Inc. All rights reserved.
摘要：
In this paper, we consider the following problem
$$
\left\{
\begin{array}{ll}
-\Delta u+u=u^{2^{*}-1}+\lambda(f(x,u)+h(x))\ \ \hbox{in}\ \mathbb{R}^{N},\\
u\in H^{1}(\mathbb{R}^{N}),\ \ u>0 \ \hbox{in}\ \mathbb{R}^{N},
\end{array}
\right. (\star)
$$
where $\lambda>0$ is a parameter, $2^* =\frac {2N}{N-2}$ is the critical Sobolev exponent and $N>4$, $f(x,t)$ and $h(x)$ are some given functions. We
prove that there exists $0<\lambda^{*}<+\infty$ such that $(\star)$ has
exactly two positive solutions for $\lambda\in(0,\lambda^{*})$ by
Barrier method and Mountain Pass Lemma and no positive solutions for $\lambda >\lambda^*$. Moreover,
if $\lambda=\lambda^*$, $(\star)$ has a unique solution $(\lambda^{*}, u_{\lambda^{*}})$, which means that $(\lambda^{*}, u_{\lambda^{*}})$ is a
turning point in $H^{1}(\mathbb{R}^{N})$ for problem $(\star)$.
