作者机构：
[Peng, Shuangjie; Li, Gongbao] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Gongbao] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
摘要：
We give some regularity results of the solutions and a Liouville type theorem to singular elliptic equations involving the Caffarelli-Kohn-Nirenberg inequalities.
期刊：
JOURNAL D ANALYSE MATHEMATIQUE,2008年104(1):125-154 ISSN：0021-7670
通讯作者：
Deng, Yinbin
作者机构：
[Shuangjie Peng; Yinbin Deng; Lingyu Jin] Huazhong Normal Univ, Dept Math, Wuhan 430070, Hubei, Peoples R China.
通讯机构：
[Deng, Yinbin] H;Huazhong Normal Univ, Dept Math, Wuhan 430070, Hubei, Peoples R China.
摘要：
Let Ω be a bounded domain with a smooth C
2 boundary in ℝn (n ≥ 3), 0 ∈
$$\bar \Omega $$
, and υ denote the unit outward normal to ∂Ω. In this paper, we are concerned with the following class of boundary value problems:
*
$$\left\{ \begin{gathered} - \Delta u - \mu \tfrac{u}{{\left| x \right|^2 }} + \lambda u = \left| u \right|^{2^* - 2} u + \eta \left| u \right|^{p - 2} u, in \Omega , \hfill \\ \tfrac{{\partial u}}{{\partial v}} + \alpha (x)u = 0, on \partial \Omega , \hfill \\ \end{gathered} \right.$$
where 2* = 2n/(n − 2) is the limiting exponent for the embedding of H
1(Ω) into L
p
(Ω), 2 < p < 2*,
$$\bar \mu \triangleq \tfrac{{(n - 2)^2 }}{4}$$
, η ≥ 0 and λ ∈ ℝ1 are parameters, and α(x) ∈ C(∂Ω), α(x) ≥ 0. Through a compactness analysis of the functional corresponding to the problem (*), we obtain the existence of positive solutions for this problem under various assumptions on the parameters μ, λ and the fact that 0 ∈ Ω or 0 ∈ ∂Ω.
作者机构：
[Kang, Dongsheng] S Cent Univ Natl, Dept Math, Wuhan 430074, Peoples R China.;[Peng, Shuangjie] Cent China Normal Univ, Sch Math & Stat, Wuhan 430071, Peoples R China.
摘要：
Let N >= 3, lambda > 0, beta <= 0, N + beta - 2 > 0, N + alpha > 0, N + sigma > 0, alpha + 2 > beta, sigma + 2 > beta, beta/2 >= sigma/p(beta,sigma), 1 < q < min(p(beta, sigma)), p(s, t) := 2(N+t)/N+s-2 be the critical Sobolev-Hardy exponent. Via the variational methods, we prove the existence of a nontrivial solution to the singular semilinear problem -div (vertical bar x vertical bar(beta)del u) = vertical bar x vertical bar(alpha) vertical bar u vertical bar(p(beta,alpha)-2)u + lambda a(x)vertical bar u vertical bar(q-2)u, u >= 0 in R-N for suitable parameters N, lambda, q and some kinds of functions a (x). (c) 2007 Elsevier Ltd. All rights reserved.
期刊：
REVISTA MATEMATICA IBEROAMERICANA,2007年23(3):1039-1066 ISSN：0213-2230
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, Sch Math Stat & Comp Sci, Armidale, NSW 2351, Australia.
通讯机构：
[Li, Gongbao] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
摘要：
We prove the existence of positive solutions concentrating simultaneously on some higher dimensional manifolds near and on the boundary of the domain for a nonlinear singularly perturbed elliptic Neumann problem.
摘要：
We study the radially symmetric Schrödinger equation
$$ - \varepsilon ^{2} \Delta u + V{\left( {|x|} \right)}u = W{\left( {|x|} \right)}u^{p} ,\quad u > 0,\;\;u \in H^{1} ({\mathbb{R}}^{N} ), $$
with N ≥ 1, ɛ > 0 and p > 1. As ɛ→ 0, we prove the existence of positive radially symmetric solutions concentrating simultaneously on k spheres. The radii are localized near non-degenerate critical points of the function
$$\Gamma (r) = r^{{N - 1}} {\left[ {V(r)} \right]}^{{\frac{{p + 1}}{{p - 1}} - \frac{1}{2}}} {\left[ {W(r)} \right]}^{{ - \frac{2} {{p - 1}}}}. $$
摘要：
We investigate elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities:
$${-div(|x|^\alpha |\nabla u|^{p-2}\nabla u)=|x|^\beta u^{p(\alpha,\beta)-1}, u(x) > 0,\,x\in\Omega\subset\mathbb{R}^N} (N\geq 3), 1<p<N$$
and
$${\alpha, \beta\in \mathbb{R}}$$
such that
$${\frac{p(N+\beta)}{N-p+\alpha} > p}$$
. For various parameters α, β and various domains Ω, we establish some existence and non-existence results of solutions in rather general, possibly degenerate or singular settings.
关键词：
We consider the existence and asymptotic behavior of standing wave solutions to nonlinear Schrödinger equations with electromagnetic fields: ih∂ψ∂t=(hi∇−A(x))2ψ+W(x)ψ−f(|ψ|2)ψ ih∂ψ∂t=(hi∇−A(x))2ψ+W(x)ψ−f(|ψ|2)ψ on R×Ω R×Ω . Ω⊂RN Ω⊂RN is a domain which may be bounded or unbounded. For h>0 h>0 small we obtain the existence of multi-bump bound states ψh(x;t)=e−iEt/huh(x) ψh(x;t)=e−iEt/huh(x) where uh uh concentrates simultaneously at possibly degenerate;non-isolated local minima of W W as h→0 h→0 . We require that W≥E W≥E and allow the possibility that {x∈Ω:W(x)=E}≠∅ {x∈Ω:W(x)=E}≠∅ . Moreover;we describe the asymptotic behavior of uh uh as h→0 h→0 . Published: 2006 First available in Project Euclid: 18 December 2012 zbMATH: 1146.35081 MathSciNet: MR2236582 Digital Object Identifier: 10.57262/ade/1355867676 Subjects: Primary: 35J60 Secondary: 35B33;35Q55
摘要：
We consider the existence and asymptotic behavior of standing wave solutions to nonlinear Schrödinger equations with electromagnetic fields: $ih\frac{\partial\psi}{\partial t} =\left(\frac{h}{i}\nabla-A(x)\right)^2\psi+W(x)\psi-f(|\psi|^2)\psi$ on ${{\mathbb R}}\times{\Omega}$. $\Omega\subset{{\mathbb R}}^N$ is a domain which may be bounded or unbounded. For $h>0$ small we obtain the existence of multi-bump bound states $\psi_h (x,t)=e^{-iEt/h}u_h(x)$ where $u_h$ concentrates simultaneously at possibly degenerate, non-isolated local minima of $W$ as $h\to0$. We require that $W\geq E$ and allow the possibility that $\{x\in\Omega:W(x)=E\}\not=\emptyset$. Moreover, we describe the asymptotic behavior of $u_h$ as $h\to\, 0$.
作者机构：
[Cao, Daomin] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;Chinese Acad Sci, Inst Appl Math, AMSS, Beijing 100080, Peoples R China.
通讯机构：
[Cao, Daomin] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Multi-bump;Standing waves;Variational method;Nonlinear Schrödinger equation;Bound states
摘要：
We consider existence and qualitative properties of standing wave solutions
$$\Psi(x,t) = e^{-iEt/h}u(x)$$
to the nonlinear Schrödinger equation
$$ih\frac{\partial\psi} {\partial t} = -\frac{h^2}{2m}\Delta\psi+W(x)\psi-|\psi|^{p-1}\psi = 0$$
with E being a critical frequency in the sense that inf
$$_{x\in\mathbb{R}^N}W(x)=E$$
. We verify that if the zero set of W − E has several isolated points x
i
(
$$i=1,\ldots,m$$
) near which W − E is almost exponentially flat with approximately the same behavior, then for h > 0 small enough, there exists, for any integer k,
$$1\leq k\leq m$$
, a standing wave solution which concentrates simultaneously on
$$\{x_j|j=1,\ldots,k\}$$
, where
$$\{x_j|j=1,\ldots,k\}$$
is any given subset of
$$\{x_i|i=1,\ldots,m\}$$
. This generalizes the result of Byeon and Wang in 3 (Arch Rat Mech Anal 165: 295–316, 2002).
摘要：
Let
$$B_R \subset R^N (N\geq 3)$$
be a ball centered at the origin with radius R. We investigate the asymptotic behavior of positive solutions for the Dirichlet problem
$$-\Delta u=\frac{\mu u}{|x|^2}+u^{2^*-1-\varepsilon}, u > 0 $$
in
$$B_R, u=0$$
on ∂BR when ɛ→+ for suitable positive numbers μ
