摘要:
In this paper, we study a kind of nonlinear Schrodinger-Poisson system with a parameter epsilon. For any positive integer m, we prove that there exists epsilon(m) > 0 such that, for 0 < epsilon < epsilon(m), the equation has an m-bump positive solution under some suitable conditions. As a consequence, the equation has more and more multi-bump positive solutions as epsilon -> 0. (C) 2011 American Institute of Physics. [doi:10.1063/1.3585657]
作者机构:
[Li, Yi] Univ Iowa, Dept Math, Iowa City, IA 52242 USA.;[Li, Yi] Xi An Jiao Tong Univ, Coll Sci, Xian 710049, Peoples R China.;[Peng, Shuangjie] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构:
[Li, Yi] U;Univ Iowa, Dept Math, Iowa City, IA 52242 USA.
摘要:
Let Omega be a bounded domain in R(N) (N >= 2), phi a harmonic function in (Omega) over bar. In this paper we study the existence of solutions to the following problem arising in the study of vortex pairs (P(lambda)) {-Delta u =lambda(u - phi)(+)(p-1), x is an element of Omega, u = 0, x is an element of partial derivative Omega. The set Omega(p) = {x is an element of Omega, u(x) > phi}phi is called "vortex core". Existence of solutions whose "vortex core" consists of one component and asymptotic behavior of "vortex core" were studied by many authors for large lambda recently. Under the condition that phi has k strictly local minimum points on the boundary partial derivative Omega, we obtain in this paper that for lambda large enough, (P(lambda)) has a solution with "vortex core" consisting of k components by a constructive way. (C) 2010 Elsevier Inc. All rights reserved.
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:
This paper is concerned with a singular elliptic system, which involves the Caffarelli-Kohn-Nirenberg inequality and multiple critical exponents. By analytic technics and variational methods, the extremals of the corresponding bet Hardy-Sobolev constant are found, the existence of positive solutions to the system is established and the asymptotic properties of solutions at the singular point are proved.
摘要:
In this paper, we study the nonlinear Schrodinger equation with electromagnetic fields (del/i - A(vertical bar y vertical bar))(2)u + V(vertical bar y vertical bar)u = vertical bar u vertical bar(p-1)u, u : R-N bar right arrow C, where the vector A(r) = (A(1)(r), A(2)(r), ... , A(N)(r)) is such that A(j)(r) (j = 1, 2, ... , N) is a real function on R+ and V(r) is a positive function on R+, 1 < p < N+2/N-2 if N >= 3 and 1 < p < +infinity if N = 2. We prove that the equation has infinitely many non-radial complex-valued solutions under conditions (H-1) and (H-2) which are given in Section 1. (C) 2011 Elsevier Inc. All rights reserved.
期刊:
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS,2010年12(6):1069-1092 ISSN:0219-1997
通讯作者:
Li, Gongbao
作者机构:
[SHUANGJIE PENG; GONGBAO LI] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[SHUSEN YAN] Univ New England, Dept Math, Armidale, NSW 2351, Australia.
通讯机构:
[Li, Gongbao] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词:
Infinitely many solutions;non-radial solutions;reduction argument;Schrödinger–Poisson system
摘要:
We consider the following nonlinear Schrodinger-Poisson system in R-3 {-Delta u + u + K(vertical bar y vertical bar)Phi(y)u = Q(vertical bar y vertical bar)vertical bar u vertical bar(p-1)u, y is an element of R-3, -Delta Phi= K(vertical bar y vertical bar)u(2), y is an element of R-3, (0.1) where K(r) and Q(r) are bounded and positive functions, 1 < p < 5. Assume that K(r) and Q(r) have the following expansions (as r -> +infinity): K(r) = a/r(m) + O (1/r(m+theta)), Q(r) = Q(0) + b/r(n) + O(1/r(n+kappa)), where a > 0, b is an element of R, m > 1/2, n > 1, theta > 0, kappa > 0, and Q(0) > 0 are some constants. We prove that (0.1) has infinitely many non-radial positive solutions if b < 0, or if b >= 0 and 2m < n.
期刊:
Advances in Mathematics,2010年225(5):2741-2785 ISSN:0001-8708
通讯作者:
Cao, Daomin
作者机构:
[Daomin Cao] Chinese Acad Sci, Inst Appl Math, Beijing 100190, Peoples R China.;[Shuangjie Peng] Cent China Normal Univ, Sch Math & Stat, Wuhan, Peoples R China.;[Shusen Yan] Univ New England, Sch Math Stat & Comp Sci, Armidale, NSW 2351, Australia.
摘要:
Let Omega be a bounded domain in R(2), u(+) = u if u >= 0, u(+) = 0 if u < 0, u(-) = u(+) - u. In this paper we study the existence of solutions to the following problem arising in the study of a simple model of a confined plasma (P(lambda)) {Delta u - lambda u(-) = o, in Omega, u = c, on partial derivative Omega, integral partial derivative u/partial derivative nu ds = I, partial derivative Omega where nu is the outward unit normal of partial derivative Omega at x, c is a constant which is unprescribed, and I is a given positive constant. The set Omega(p) = {x is an element of Omega, u(x) < 0} is called plasma set. Existence of solutions whose plasma set consisting of one component and asymptotic behavior of plasma set were studied by Caffarelli and Friedman (1980) [3] for large lambda. Under the condition that the homology of Omega is nontrivial we obtain in this paper by a constructive way that for any given integer k >= 1, there is lambda(k) > 0 such that for lambda > lambda(k), (P(lambda)) has a solution with plasma set consisting of k components. (C) 2010 Elsevier Inc. All rights reserved.
摘要:
We construct spike layered solutions for the semilinear elliptic equation -epsilon(2)Delta u + V (x)u = K(x)u(p-1) on a domain Omega subset of R(N) which may be bounded or unbounded. The solutions concentrate simultaneously on a finite number of m-dimensional spheres in Omega. These spheres accumulate as epsilon -> 0 at a prescribed sphere in Omega whose location is determined by the potential functions V, K. (c) 2010 Elsevier Inc. All rights reserved.
期刊:
Communications in Partial Differential Equations,2009年34(12):1566-1591 ISSN:0360-5302
通讯作者:
Peng, Shuangjie
作者机构:
[Daomin Cao] Chinese Acad Sci, Inst Appl Math, AMSS, Beijing, Peoples R China.;[Shuangjie Peng] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构:
[Peng, Shuangjie] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词:
Bound states;Concentrating solutions;Critical point;Potential vanishing or unbounded at infinity;Variational method
摘要:
In this paper we study the existence and qualitative property of standing wave solutions for the nonlinear Schrödinger equation . Let . For any integer k ≥ 1, we prove existence of standing wave solutions with u > 0 having k local maximum points and concentrating near a given local maximum point of Γ when ε is small. The potentials V and K are allowed to be either vanishing or unbounded at infinity. Existence of solutions concentrating near k distinct non-degenerate critical points of Γ has been proved under the same assumptions on V and K as well.
作者机构:
[Shuang-jie; Kou] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Kou] PLA Univ Sci & Technol, Inst Sci, Nanjing 210007, Peoples R China.
通讯机构:
[Kou, Bing-yu] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词:
Singular equations;Caffarelli-Kohn-Nirenberg inequalities;critical exponents;ground state sloutions
摘要:
Let Omega be a bounded domain with a smooth C(2) boundary in R(N)(N >= 3), 0 is an element of (Omega) over bar, and n denote the unit outward normal to partial derivative Omega. We are concerned with the Neumann boundary problems: -div(vertical bar x vertical bar(alpha)vertical bar del u vertical bar(p-2)del u) = vertical bar x vertical bar(beta)u(p(alpha,beta)-1) - lambda vertical bar x vertical bar(gamma)u(p-1), u(x) > 0, x is an element of Omega, partial derivative u/partial derivative n = 0 on partial derivative Omega, where 1 < p < N and alpha < 0, beta < 0 such that p(alpha, beta) (sic) p(N+beta)/N-p+alpha > p, gamma > alpha-p. For various parameters alpha, beta or gamma, we establish certain existence results of the solutions in the case 0 is an element of Omega or 0 is an element of partial derivative Omega.
摘要:
The main purpose of this paper is to analyze the asymptotic behavior of the radial solution of Hénon equation −Δu = |x|
α
u
p−1, u > 0, x ∈ B
R
(0) ⊂ ℝ
n
(n ⩾ 3), u = 0, x ∈ ∂B
R
(0), where
$$
p \to p(\alpha ) = \frac{{2(n + \alpha )}}
{{n - 2}}
$$
from left side, α > 0.
作者机构:
[Cao, Daomin] Chinese Acad Sci, Inst Appl Math, Beijing 100190, Peoples R China.;[Peng, Shuangjie] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Yan, Shusen] Univ New England, Sch Math Stat & Comp Sci, 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:
Asymptotic behaviour;Ground state solutions;Hénon equation
期刊:
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS,2009年11(2):185-199 ISSN:0219-1997
通讯作者:
Deng, Yinbin
作者机构:
[SHUANGJIE PENG; YINBIN DENG] Huazhong Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[LINGYU JIN] S China Agr Univ, Coll Sci, Guangzhou 510642, Guangdong, Peoples R China.
通讯机构:
[Deng, Yinbin] H;Huazhong Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词:
Caffarelli–Kohn–Nirenberg inequalities;degeneracy;singularity;critical point
摘要:
In this paper, we are concerned with the following elliptic problems which are related to the well-known Caffarelli-Kohn-Nirenberg inequalities: -div(vertical bar x vertical bar-(2a)del u) - lambda vertical bar x vertical bar(-eE)u = vertical bar x vertical bar(-bp)vertical bar u vertical bar(p-2)u + eta vertical bar x vertical bar(-dD)vertical bar u vertical bar(q-2)u in Omega u = 0 on partial derivative Omega, (0.1) where a = b < 0,p = 2N/N-2 (N >= 3), a <= d <= a + 1, a <= e <= a +1, D = 2N/N-2-2(a-d), E = 2N/N-2-2(a-e), 2 < q < D, lambda and eta are real constants. We obtain positive solutions for problem (0.1). Moreover, we establish a corresponding Pohozaev identity for problem (0.1), from which, the nonexistence of positive solutions for problem ( 0.1) is obtained.
作者机构:
[Dao-min Cao] Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China.;[Shuang-jie Peng] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构:
[Cao, Dao-min] C;Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China.
关键词:
Mean curvature;critical point;concentrating solutions
摘要:
We consider the singularly perturbed equation (Formula Presented) on a domain Ω ∈ ℝN which may be bounded or unbounded. Under suitable hypotheses on V, K we construct layered solutions u μ H 01 (Omega) which concentrate on certain high-dimensional subsets of Ω. This gives a positive answer to a problem proposed by Ambrosetti, Malchiodi and Ni in [1].