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1.
Let M be a smooth, compact (closed and without boundary) surface. Consider the metricsσ(z)|dz|~2 and ρ(ω)|dω|~2 on M, where z=x+iy and ω=u+iv are conformal coordinates onM. For a Lipschitz map ω=ω(z): (M, σ|dz|~2)→(M, ρ|dw|~2), we define the energy density of 相似文献
2.
主要考虑如下加权全能量极值问题:h∈■^(inf)(A_(1),A_(2))α∫∫A_(1)(|h_(N)|^(2)+|h_(T)|^(2))1/(|h_(z)|^(2))dz+β∫∫A_(1)|h_(N)|^(2)+|h_(T)|^(2)/J(z,h)1/|z|^(2)dz,其中■(A_(1),A_(2))代表从圆环A_(1)到圆环A_(2)的所有保向同胚映射的集合.研究得到唯一的极值映射为径向拉伸映射.这将[Iwaniec T,Onninen J.Hyperelastic deformations of smallest total energy[J].Arch Rational Mech Anal,2009,194:927-986.]的结果推广至非欧情形.同时,也分别研究了圆环上的加权调和能量的极值问题与加权偏差的极值问题. 相似文献
3.
4.
本文考虑临界耦合的Hartree方程组{-△+λu=∫Ω|u(z)|^2*μ/|x-z|μdz|u|^2*μ-2u+βν,x∈Ω,-△+νu=∫Ω|ν(z)|^2*μ/|x-z|μdz|u|^2*μ-2u+βν,x∈Ω,其中Ω是RN中带有光滑边界的有界区域,N≥3,λ,v是常数,且满足λ,v>-λ1(Ω),λ1(Ω)是(-△,H01(Ω))的第一特征值,β> 0是耦合参数,临界指标2μ*=(2N-μ)/(N-2)来源于Hardy-LittlewoodSobolev不等式,利用变分的方法证明了临界Hartree方程组基态正解的存在性. 相似文献
5.
首先证明广义Bergman空间A_(N,α)~p,(α-n-1,p0)上的复合算子C_φ的有界性和紧性是不依赖于p的,进而证明了若对某个q0和-n-1βα,C_φ在A_(N,β)~α上有界,则C_φ在A_(N,α)~p,α(α-n-1,p0)上是紧的当且仅当lim|z|→1-1-(|z|~2/1-|φ(1)|~2)=0. 相似文献
6.
在一定条件下,证明了(?)~n中单位球上的加权Bergman空间A~p(φ)上的复合算子C_φ是紧算子的充要条件是当|z|→1~-时(1-|x|~2)/(1-|φ(z)|~2)→0. 相似文献
7.
Let M be a smooth,compact(closed and without boundary)surface.Consider the metrics σ(z)|dz|2 and ρ(ω)|dω|2 on M,where z=x+iy and ω=u+iv are conformal coordinates on M. 相似文献
8.
9.
孙小妹 《数学物理学报(A辑)》2018,(1)
该文研究如下形式的Choquard型方程-△_pu+V(x)|u|~(p-2)u=(|x|~(-(N-α))*F(u))f(u),其中,-△_pu=div(|▽u|~(p-2)▽u)),x=(y,z)∈R~K×R~(N-K).假定混合位势V(y,z)关于y具有周期性,关于z具有强制性,并且非线性项f满足一定的条件,利用变分理论,该文证明了上述Choquard型方程具有山路水平解. 相似文献
10.
设0
-1,D_α~p表示单位球B_n上满足∫_(B_n)(1-|z|~2)~α|▽f(z)|~pdv(z)<∞的全纯函数全体.对0
相似文献
11.
Existence and concentration result for Kirchhoff equations with critical exponent and Hartree nonlinearity
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This paper is concerned with the following Kirchhoff-type equations
$$
\left\{
\begin{array}{ll}
\displaystyle
-\big(\varepsilon^{2}a+\varepsilon b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\mathrm{d}x\big)\Delta u
+ V(x)u+\mu\phi |u|^{p-2}u=f(x,u), &\quad \mbox{ in }\mathbb{R}^{3},\(-\Delta)^{\frac{\alpha}{2}} \phi=\mu|u|^{p},~u>0, &\quad \mbox{ in }\mathbb{R}^{3},\\end{array}
\right.
$$
where $f(x,u)=\lambda K(x)|u|^{q-2}u+Q(x)|u|^{4}u$, $a>0,~b,~\mu\geq0$ are constants, $\alpha\in(0,3)$, $p\in[2,3),~q\in[2p,6)$ and $\varepsilon,~\lambda>0$ are parameters. Under some mild conditions on $V(x),~K(x)$ and $Q(x)$, we prove that the above system possesses a ground state solution $u_{\varepsilon}$ with exponential decay at infinity for $\lambda>0$ and $\varepsilon$ small enough. Furthermore, $u_{\varepsilon}$ concentrates around a global minimum point of $V(x)$ as $\varepsilon\rightarrow0$. The methods used here are based on minimax theorems and the concentration-compactness principle of Lions. Our results generalize and improve those in Liu and Guo (Z Angew Math Phys 66: 747-769, 2015), Zhao and Zhao (Nonlinear Anal 70: 2150-2164, 2009) and some other related literature. 相似文献
12.
Hu Ke 《数学年刊B辑(英文版)》1983,4(2):187-190
AIn this paper, the author obtains the following results:(1) If Taylor coeffiients of a function satisfy the conditions:(i),(ii),(iii)A_k=O(1/k) the for any h>0 the function φ(z)=exp{w(z)} satisfies the asymptotic equality the case h>1/2 was proved by Milin.(2) If f(z)=z α_2z~2 …∈S~* and,then for λ>1/2 相似文献
13.
Ismail Kombe 《Proceedings of the American Mathematical Society》2004,132(9):2683-2691
In this paper we consider the following initial value problem:
where and . Nonexistence of positive solutions is analyzed.
where and . Nonexistence of positive solutions is analyzed.
14.
Philippe Poulin 《Proceedings of the American Mathematical Society》2007,135(1):77-85
It is well known that the Green function of the standard discrete Laplacian on , exhibits a pathological behavior in dimension . In particular, the estimate fails for . This fact complicates the study of the scattering theory of discrete Schrödinger operators. Molchanov and Vainberg suggested the following alternative to the standard discrete Laplacian, and conjectured that the estimate holds for all . In this paper we prove this conjecture.
15.
An improved Hardy-Sobolev inequality and its application 总被引:4,自引:0,他引:4
Adimurthi Nirmalendu Chaudhuri Mythily Ramaswamy 《Proceedings of the American Mathematical Society》2002,130(2):489-505
For , a bounded domain, and for , we improve the Hardy-Sobolev inequality by adding a term with a singular weight of the type . We show that this weight function is optimal in the sense that the inequality fails for any other weight function more singular than this one. Moreover, we show that a series of finite terms can be added to improve the Hardy-Sobolev inequality, which answers a question of Brezis-Vazquez. Finally, we use this result to analyze the behaviour of the first eigenvalue of the operator as increases to for .
16.
设$\mu$是$[0,1)$上的正规函数,
给出了${\bf C}^{\it n}$中单位球$B$上$\mu$-Bloch空间$\beta_{\mu}$中函数的几种刻画. 证明了下列条件是等价的:
(1) $f\in \beta_{\mu}$; \
(2) $f\in H(B)$且函数$\mu(|z|)(1-|z|^{2})^{\gamma-1}R^{\alpha,\gamma}f(z)$ 在$B$上有界;
(3) $f\in H(B)$ 且函数${\mu(|z|)(1-|z|^{2})^{M_{1}-1}\frac{\partial^{M_{1}} f}{\partial z^{m}}(z)}$ 在$B$上有界, 其中$|m|=M_{1}$;
(4) $f\in H(B)$ 且函数${\mu(|z|)(1-|z|^{2})^{M_{2}-1}R^{(M_{2})}f(z)}$ 在$B$上有界. 相似文献
17.
A global compactness result for singular elliptic problems involving critical Sobolev exponent 总被引:11,自引:0,他引:11
Let be a bounded domain such that . Let be a (P.S.) sequence of the functional . We study the limit behaviour of and obtain a global compactness result.
18.
王建飞 《数学年刊A辑(中文版)》2013,34(2):223-234
在有界星形圆形域上定义了一个新的星形映射子族, 它包含了$\alpha$阶星形映射族和$\alpha$阶强星形映射族作为两个特殊子类.
给出了此类星形映射子族的增长定理和掩盖定理. 另外, 还证明了Reinhardt域$\Omega_{n,p_{2},\cdots,p_{n}}$上此星形映射子族在Roper-Suffridge算子
\begin{align*}
F(z)=\Big(f(z_{1}),\Big(\frac{f(z_{1})}{z_{1}}\Big)^{\beta_{2}}(f'(z_{1}))^{\gamma_{2}}z_{2},\cdots,
\Big(\frac{f(z_{1})}{z_{1}}\Big)^{\beta_{n}}(f'(z_{1}))^{\gamma_{n}}z_{n}\Big)'
\end{align*}
作用下保持不变, 其中
$\Omega_{n,p_{2},\cdots,p_{n}}=\{z\in
{\mathbb{C}}^{n}:|z_1|^2+|z_2|^{p_2}+\cdots + |z_n|^{p_n}<1\}$,
$p_{j}\geq1$, $\beta_{j}\in$ $[0, 1]$, $\gamma_{j}\in[0,
\frac{1}{p_{j}}]$满足$\beta_{j}+\gamma_{j}\leq1$,
所取的单值解析分支使得 $\big({\frac{f(z_{1})}{z_{1}}}\big)^{\beta_{j}}\big|_{z_{1}=0}=1$,
$(f'(z_{1}))^{\gamma_{j}}\mid_{{z_{1}=0}}=1$, $j=2,\cdots,n$. 这些结果不仅包含了许多已有的结果, 而且得到了新的结论. 相似文献
19.
Markus Kunze 《Proceedings of the American Mathematical Society》2003,131(7):2181-2188
We consider a non-local variational problem whose critical points are related to bound states in certain optical fibers. The functional is given by , and relying on the regularizing properties of the solution to the free Schrödinger equation, it will be shown that has infinitely many critical points.
20.
Hu Ke 《数学年刊B辑(英文版)》1980,1(34):421-427
Let \[f(z) = z + \sum\limits_{n = 1}^\infty {{a_n}{z^n} \in S} {\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {\kern 1pt} \log \frac{{f(z) - f(\xi )}}{{z - \xi }} - \frac{{z\xi }}{{f(z)f(\xi )}} = \sum\limits_{m,n = 1}^\infty {{d_{m,n}}{z^m}{\xi ^n},} \], we denote \[{f_v} = f({z_v})\] , \[\begin{array}{l}
{\varphi _\varepsilon }({z_u}{z_v}) = {\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}} \right|^\varepsilon }\frac{1}{{(1 - {z_u}{{\bar z}_v})}},\g_m^\varepsilon (z) = - {F_m}(\frac{1}{{f(z)}}) + \frac{1}{{{z^m}}} + \varepsilon {{\bar z}^m},
\end{array}\], where \({F_m}(t)\) is a Faber polynomial of degree m.
Theorem 1. If \[f(z) \in S{\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{u,v = 1}^N {{A_{u,v}}{x_u}{{\bar x}_v} \ge 0} \] and then \[\begin{array}{l}
\sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} {\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}} \right|^\varepsilon }\exp \{ \alpha {F_l}({z_u},{z_v})\} \ \le \sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} \varphi _\varepsilon ^\alpha ({z_u}{z_v})l = 1,2,3,
\end{array}\], where \[\begin{array}{l}
{F_1}({z_u},{z_v}) = \frac{1}{2}\sum\limits_{n = 1}^\infty {\frac{1}{n}} g_n^\varepsilon ({z_u})\bar g_n^\varepsilon ({z_v}),\{F_2}({z_u},{z_v}) = \frac{1}{{1 + {\varepsilon _n}R{d_{n,n}}}}Rg_n^\varepsilon ({z_u})Rg_n^\varepsilon ({z_v}),\{F_3}({z_u},{z_v}) = \frac{1}{{1 - {\varepsilon _n}R{d_{n,n}}}}Rg_n^\varepsilon ({z_u})Rg_n^\varepsilon ({z_v}).
\end{array}\] The \[F({z_u},{z_v}) = \frac{1}{2}{g_1}({z_u}){{\bar g}_2}({z_v})\] is due to Kungsun.
Theorem 2. If \(f(z) \in S\) ,then \[P(z) + \left| {\sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} {{\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}\frac{{{z_u}{z_v}}}{{{f_u}{f_v}}}} \right|}^\varepsilon }} \right| \le \sum\limits_{u,v = 1}^N {{\lambda _u}{{\bar \lambda }_v}} \frac{1}{{1 - {z_u}{{\bar z}_v}}}\], where \[\begin{array}{l}
P(z) = \frac{1}{2}\sum\limits_{n = 1}^\infty {\frac{1}{n}} {G_n}(z),\{G_n}(z) = {\left| {\left| {\sum\limits_{n = 1}^N {{\beta _u}({F_n}(\frac{1}{{f({z_u})}}) - \frac{1}{{z_u^n}})} } \right| - \left| {\sum\limits_{n = 1}^N {{\beta _u}z_u^n} } \right|} \right|^2},
\end{array}\], \(P(z) \equiv 0\) is due to Xia Daoxing. 相似文献