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