Remarks on a Mean Field Equation on $\mathbb{S}^2$

In this note, we study symmetry of solutions of the elliptic equation\begin{equation*} -\Delta _{\mathbb{S}^{2}}u+3=e^{2u}\ \ \hbox{on}\ \ \mathbb{S}^{2},\end{equation*} that arises in the consideration of rigidity problem of Hawking mass in general relativity. We provide various conditions under which this equation has only constant solutions, and consequently imply the rigidity of Hawking mass for stable constant mean curvature (CMC) sphere.     

$L^{p}_{[0,1]}$空间M\"{u}ntz有理逼近的Jackson型估计 (英)

设$\Lambda=\{\lambda_{n}\}_{n=1}^{\infty}$为正的实数数列, 且当$n\rightarrow\infty$时, 有$\lambda_{n}\searrow 0$.本文给出了当 $\lambda_{n}\leq Mn^{-\frac{1}{2}},\;n=1,2, \cdots ,$(其中$M>0$为一正常数)时M\"{u}ntz系统$\{x^{\lambda_n}\}$的有理函数在$ L_{[0,1]} ^{p}$空间的逼近速度,主要结论为$R_{n} (f, \Lambda )_{L^{p}}\leq C_M \omega (f, n^{-\frac{1}{2}})_{L^{p}},\;1 \leq p \leq \infty.$     

由H\"{o}rmander向量场构成的抛物方程的 $W_{\ast}^{1,p}$ 正则性

设 $X_{1},\cdots ,X_{q}\ (q相似文献   

Regularity of Positive Solutions for an Integral System on Heisenberg Group

In this paper, we are concerned with the properties of positive solutions of the following nonlinear integral systems on the Heisenberg group $\mathbb{H}^n$, \begin{equation} \left\{\begin{array}{ll} u(x)=\int_{\mathbb{H}^n}\frac{v^{q}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ v(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ w(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)v^{q}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ \end{array}\right.\end{equation} for $x\in \mathbb{H}^n$, where $0<\alpha 1$ satisfying $\frac{1}{p+1} $+ $\frac{1}{q+1} + \frac{1}{r+1} = \frac{Q+α+β}{Q}.$ We show that positive solution triples $(u,v,w)\in L^{p+1}(\mathbb{H}^n)\times L^{q+1}(\mathbb{H}^n)\times L^{r+1}(\mathbb{H}^n)$ are bounded and they converge to zero when $|x|→∞.$     

Multiple positive solutions of singular and critical elliptic problem in {\mathbb{R}^2} with discontinuous nonlinearities

Let Ω be a bounded domain in ${\mathbb{R}^2}$ with smooth boundary. We consider the following singular and critical elliptic problem with discontinuous nonlinearity: $$(P_\lambda)\left \{\begin{array}{ll} - \Delta u = \lambda \left(\frac{m(x, u) e^{\alpha{u}^2}}{|x|^{\beta}} + u^{q}g(u - a)\right),\quad{u} > 0 \quad {\rm in} \quad \Omega\\u \quad \quad = 0\quad {\rm on} \quad \partial \Omega \end{array}\right.$$ where ${0\leq q < 1 ,0< \alpha\leq4\pi}$ and ${\beta \in [0, 2)}$ such that ${\frac{\beta}{2} + \frac{\alpha}{4\pi} \leq 1}$ and ${{g(t - a) = \left\{\begin{array}{ll}1, t \leq a\\ 0, t > a.\end{array}\right.}}$ Under the suitable assumptions on m(x, t) we show the existence and multiplicity of solutions for maximal interval for λ.     

Infinitely many radial solutions of a variational problem related to dispersion-managed optical fibers

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.

     

带粗糙初始值向列型液晶流的适定性

考虑了R~n上n(n≥2)维向列型液晶流(u,d)当初值属于Q_α~(-1)(R~n,R~n)×Q_α(R~n,S~2)(其中α∈(0,1))时Cauchy问题的适定性,这里的Q_α(R~n)最早由Essen,Janson,Peng和Xiao(见[Essen M,Janson S,Peng L,Xiao J.Q space of several real variables,Indiana Univ Math J,2000,49:575-615])引入,是指由R~n中满足的所有可测函数f全体所组成的空间.上式左端在取遍Rn中所有以l(I)为边长且边平行于坐标轴的立方体I的全体中取上确界,而Q_α~(-1)(R~n):=▽·Q_α(R~n).最后证明了解(u,d)在类C([0,T);Q_(α,T)~(-1)(R~n,R~n))∩L_(loc)~∞((0,T);L~∞(R~n,R~n))×C([0,T);Q_α,T(R~n,S~2))∩L_(loc)~∞((0,T);W~(1,∞)(R~n,S~2))(其中0T≤∞)中是唯一的.     

Remarks on the regularity criteria of the solutions of the 3D micropolar fluid equations

We provide two regularity criteria for the weak solutions of the 3D micropolar fluid equations, the first one in terms of one directional derivative of the velocity, i.e., $\partial_{3}u$, while the second one is is in terms of the behavior of the direction of the velocity $\frac{u}{|u|}$. More precisely, we prove that if \begin{equation*} \partial_{3}u \in L^{\beta}(0,T;L^{\alpha}(\mathbb{R}^{3}))\quad\text{ with }\frac{2}{\beta}+\frac{3}{\alpha}\leq 1+\frac{1}{\alpha}, 2&lt; \alpha \leq\infty, 2\leq\beta&lt; \infty; \end{equation*} or \begin{equation*} \operatorname{div}\left(\frac{u}{|u|}\right)\in L^{\frac{4}{1-2r}}(0,T;\dot{X}_{r}(\mathbb{R}^{3}))\quad \text{ with } 0\leq r&lt; \frac{1}{2}, \end{equation*} then the weak solution $(u(x,t),\omega(x,t))$ is regular on $\mathbb{R}^{3}\times [0,T]$. Here $\dot{X}_{r}(\mathbb{R}^{3})$ is the multiplier space.     

On the existence of full dimensional KAM torus for fractional nonlinear Schrodinger equation

In this paper,\ we study fractional nonlinear Schrodinger equation (FNLS) with periodic boundary condition $$ \textbf{i}u_{t}=-(-\Delta)^{s_{0}} u-V*u-\epsilon f(x)|u|^4u,\ ~~x\in \mathbb{T}, ~~t\in \mathbb{R}, ~~s_{0}\in (\frac12,1),~~~~~~~~~~~~~~~~~~~~~~~~~~~~(0.1) $$ where $(-\Delta)^{s_{0}}$ is the Riesz fractional differentiation defined in [21] and $V*$ is the Fourier multiplier defined by $\widehat{V*u}(n)=V_n\widehat{u}(n),\ V_n\in\left[-1,1\right],$ and $f(x)$ is Gevrey smooth. We prove that for $0\leq|\epsilon|\ll1$ and appropriate $V$,\ the equation (0.1) admits a full dimensional KAM torus in the Gevrey space satisfying $ \frac12e^{-rn^{\theta}}\leq \left|q_n\right|\leq 2e^{-rn^{\theta}}, \theta\in (0,1),$ which generalizes the results given by [8-10] to fractional nonlinear Schrodinger equation.     

Multiple sign-changing solutions for a class of semilinear elliptic equations in $\mathbb{R}^{N}$

In this paper, we study the following semilinear elliptic equations $$-\triangle u+V(x)u=f(x,u), \ \ x\in \mathbb{R}^{N},$$ where $V\in C(\mathbb{R}^{N}, \mathbb{R})$ and $f\in C(\mathbb{R}^{N}\times\mathbb{R}, \mathbb{R})$. Under some suitable conditions, we prove that the equation has three solutions of mountain pass type: one positive, one negative, and sign-changing. Furthermore, if $f$ is odd with respect to its second variable, this problem has infinitely many sign-changing solutions.     

Nonradial solutions for a conformally invariant fourth order equation in \mathbb {R}^4

We consider the following Liouville equation in

Uniqueness and existence of solutions for a singular system with nonlocal operator via perturbation method

In this work, we investigate the existence and the uniqueness of solutions for the nonlocal elliptic system involving a singular nonlinearity as follows: $$ \left\{\begin{array}{ll} (-\Delta_p)^su = a(x)|u|^{q-2}u +\frac{1-\alpha}{2-\alpha-\beta} c(x)|u|^{-\alpha}|v|^{1-\beta}, \quad \text{in }\Omega,\ (-\Delta_p)^s v= b(x)|v|^{q-2}v +\frac{1-\beta}{2-\alpha-\beta} c(x)|u|^{1-\alpha}|v|^{-\beta}, \quad \text{in }\Omega,\ u=v = 0 ,\;\;\mbox{ in }\,\mathbb{R}^N\setminus\Omega, \end{array} \right. $$ where $\Omega $ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary, $0<\alpha <1,$ $0<\beta <1,$ $2-\alpha -\beta 相似文献   

Sharp Subcritical and Critical Trudinger-Moser Inequalities on $\mathbb {R}^{2}$ and their Extremal Functions

In this paper, we study on \(\mathbb {R}^{2}\) some new types of the sharp subcritical and critical Trudinger-Moser inequality that have close connections to the study of the optimizers for the classical Trudinger-Moser inequalities. For instance, one of our results can be read as follows: Let 0 ≤ β < 2, p ≥ 0, α ≥ 0. Then

$$\sup_{\left\Vert \nabla u\right\Vert_{2}^{2}+\left\Vert u\right\Vert_{2} ^{2}\leq1}\left\Vert u\right\Vert_{2}^{p}{\int}_{\mathbb{R}^{2}}\exp\left( \alpha\left( 1-\frac{\beta}{2}\right) \left\vert u\right\vert^{2}\right) \left\vert u\right\vert^{2}\frac{dx}{\left\vert x\right\vert^{\beta}}<\infty $$

if and only if α < 4π or α = 4π, p ≥ 2. The attainability and inattainability of these sharp inequalties will be also investigated using a new approach, namely the relations between the supremums of the sharp subcritical and critical ones. This new method will enable us to compute explicitly the supremums of the subcritical Trudinger-Moser inequalities in some special cases. Also, a version of Concentration-compactness principle in the spirit of Lions ( Lions, I. Rev. Mat. Iberoam. 1(1) 145–01 1985) will also be studied.

     

Exact multiplicity for the perturbed Q-curvature problem in {\mathbb{R}^N,N \geq 5}

Let N ≥ 5 and \({{\mathcal{D}}^{2,2} (\mathbb{R}^N)}\) denote the closure of \({C_0^\infty (\mathbb{R}^N)}\) in the norm \({\|u\|_{{\mathcal{D}}^{2,2} (\mathbb{R}^N)}^2 := \int\nolimits_{\mathbb{R}^N} |\Delta u|^2.}\) Let \({K \in C^2 (\mathbb{R}^N).}\) We consider the following problem for ? ≥ 0: $$(P_\varepsilon) \left\{\begin{array}{llll}{\rm Find} \, u \in {\mathcal{D}}^{2, 2} (\mathbb{R}^N) \, \, {\rm solving} :\\ \left.\begin{array}{lll}\Delta^2 u = (1+ \varepsilon K (x)) u^{\frac{N+4}{N-4}}\\ u > 0 \end{array}\right\}{\rm in} \, \mathbb{R}^N.\end{array}\right.$$ We show an exact multiplicity result for (P _? ) for all small ? > 0.     

The linear heat equation with highly oscillating potential

In this paper we consider the following initial value problem:

where and . Nonexistence of positive solutions is analyzed.

     

Topological degree for solutions of fourth order mean field equations

We consider the following fourth order mean field equation with Navier boundary condition $$\Delta^2 u = \rho \frac{h(x) e^{u}}{\int_\Omega h e^{u}}\,\,{\rm in}\, \Omega,{\quad}u = \Delta u = 0\,\,{\rm on}\,\partial \Omega,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(*)$$ where h is a C ^2,?? positive function, ?? is a bounded and smooth domain in ${\mathbb{R}^4}$ . We prove that for ${\rho \in (32m\sigma_3, 32(m + 1)\sigma_3)}$ the degree-counting formula for (*) is given by $$d(\rho)=\left\{\begin{array}{ll}\frac{1}{m!} (-\chi (\Omega) +1) \cdot\cdot \cdot (-\chi(\Omega)+m) & {\rm for}\, m >0 ,\\ 1 & {\rm for}\, m=0\end{array}\right.$$ where ??(??) is the Euler characteristic of ??. Similar result is also proved for the corresponding Dirichlet problem $$\Delta^2 u = \rho \frac{h(x) e^{u}}{\int_\Omega h e^{u}}\quad{\rm in}\,\Omega, \quad u = \nabla u = 0 \quad {\rm on}\,\,\partial \Omega.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(**)$$      

On a Right Inverse of a Polynomial of the Laplace in the Weighted Hilbert Space $L^2 (\mathbb{R}^n,e^{−|x|^2} )$

Let $P(∆)$ be a polynomial of the Laplace operator $$∆ = \sum\limits^n_{j=1}\frac{∂^2}{∂x^2_j} \ \ on \ \ \mathbb{R}^n.$$ We prove the existence of a bounded right inverse of the differential operator $P(∆)$ in the weighted Hilbert space with the Gaussian measure, i.e., $L^2(\mathbb{R}^n ,e^{−|x|^2}).$     

$\mathbb{C}^n$中一类星形映射子族的增长定理及推广的Roper-Suffridge算子

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.     

Uniqueness of positive solutions for singular problems involving the -Laplacian

We study existence and uniqueness of positive eigenfunctions for the singular eigenvalue problem: on a bounded smooth domain with zero boundary condition. We also characterize all positive solutions of in .

     

{W^{1,1}_0} -solutions for elliptic problems having gradient quadratic lower order terms

In this paper we deal with solutions of problems of the type $$\left\{\begin{array}{ll}-{\rm div} \Big(\frac{a(x)Du}{(1+|u|)^2} \Big)+u = \frac{b(x)|Du|^2}{(1+|u|)^3} +f \quad &{\rm in} \, \Omega,\\ u=0 &{\rm on} \partial \, \Omega, \end{array} \right.$$ where ${0 < \alpha \leq a(x) \leq \beta, |b(x)| \leq \gamma, \gamma > 0, f \in L^2 (\Omega)}$ and Ω is a bounded subset of ${\mathbb{R}^N}$ with N ≥ 3. We prove the existence of at least one solution for such a problem in the space ${W_{0}^{1, 1}(\Omega) \cap L^{2}(\Omega)}$ if the size of the lower order term satisfies a smallness condition when compared with the principal part of the operator. This kind of problems naturally appears when one looks for positive minima of a functional whose model is: $$J (v) = \frac{\alpha}{2} \int_{\Omega}\frac{|D v|^2}{(1 + |v|)^{2}} + \frac{12}{\int_{\Omega}|v|^2} - \int_{\Omega}f\,v , \quad f \in L^2(\Omega),$$ where in this case a(x) ≡ b(x) = α > 0.