关于Fujita型反应扩散方程组的Cauchy问题

本文研究Ｆｕｊｉｔａ型反应扩散方程组ｕｔ－Δｕ＝α１｜ｕ｜ｑ１－１ｕ＋β１｜ｖ｜ｐ１－１ｖ，（ｘ∈ＲＮ，ｔ＞０），ｖｔ－Δｖ＝α２｜ｕ｜ｑ２－１ｕ＋β２｜ｖ｜ｐ２－１ｖ，ｕ（ｘ，０）＝ｕ０（ｘ）０，ｖ（ｘ，０）＝ｖ０（ｘ）０，（ｘ∈ＲＮ）Ｌｐ解的整体存在性和有限时间Ｂｌｏｗ ｕｐ问题．这里ｑｉ＞１，ｐｉ＞１（ｉ＝１，２），α１０，α２＞０，β１＞０，β２０，１ｐ＋∞．     

一类具有双重奇性的抛物型方程的整体解和有限熄灭

本文讨论具有双重奇性的抛物型方程ｕｔ＝ ｄｉｖ（｜△ｕ~ａ｜ｐ~－２△ｕ~ａ）,（ｘ,ｔ） ∈ Ｒ~ｎ ×（０,∞）,其中Ｐ＞ １,ａ＞ ０,ｎ≤ ２．证明当１＜ Ｐ＜ｎ（ａ＋１）／（ａｎ＋１）时,存在整体自相似解ｕｇｓ（·,ｔ） ∈ Ｌ~ｑ（Ｒ~ｎ）（ｑ＞ｓ＝~△ｎ［１－ａ（ｐ－１）］／ｐ）,但是ｕｇｓ∈~／Ｌ~ｓ（Ｒ~ｎ）（定理２．１）;同时存在有限熄灭的自相似解ｕｌｓ满足相同的积分条件（定理 ３．１）．     

具临界指数和临界非线性边界条件的拟线性椭圆型方程Neumann问题

本文给出ＲＮ（Ｎ３）中有界光滑区域Ω上的拟线性椭圆型方程：－∑Ｎｉ＝１ｘｉ·｜Ｄｕ｜ｐ－２ｕｘｉ＝λ｜ｕ｜ｐ－２ｕ＋ａ（ｘ）｜ｕ｜ｐ－２ｕ＋ｆ（ｘ，ｕ），ｘ∈Ω（λ＞０，ｐ＝Ｎｐ／（Ｎ－ｐ），２ｐ＜Ｎ）在边界条件：－｜Ｄｕ｜ｐ－２Ｄνｕ｜Ω＝ψ（ｘ）｜ｕ｜ｑ－２ｕ（ｑ＝（Ｎ－１）ｐ／（Ｎ－ｐ））下的多解性结果．     

半无穷直线上的非线性薛定谔方程

本文研究非线性薛定鄂方程的初始值和边界值问题 ｉｕ_ｔ＝ｕ_（ｘｘ）－ｇ｜ｕ｜~（ｐ－１）ｕ。０＜ｘ,ｔ＜∞,这里 ｇ＞ ０, ｐ＞ ３; ｕ（ｘ,０）＝ ｈ（ｘ）．假设 ｈ（ｘ）∈ Ｈ（ＩＲ~＋）, Ｑ（ｔ）,Ｒ（ｔ） Ｅ Ｃ（ＩＲ~＋）．对于二类不同的边界值（狄里克莱型ｕ（０,ｔ）＝Ｑ（ｔ）和鲁宾型ｕ_ｘ（０,ｔ）＋ａｕ（０,ｔ）＝Ｒ（ｔ）;这里ａ是实数）本文证明古典解。 ｕ∈ Ｃ~１（Ｌ~２）∩ Ｌ~２（Ｈ~２）的存在性,唯一性和全局性．     

关于u_t=div(｜u｜~(p-2)u)方程解的梯度Hlder连续性的一个注记(英文)

对文献［１－３］中的结果：ｕｔ＝ｄｉｖ（｜ｕ｜ｐ－２ｕ）在ΩＴ＝Ω×（０，Ｔ）上弱解的空间梯度是Ｈｏ¨ｌｄｅｒ连续的做一个补充．在这个注记里，讨论了条件ｐ＞ｍａｘ｛１，２ＮＮ＋２｝是怎样由ｕ的性质所决定的．属于ＬＮｌｏｃ（ΩＴ）空间解的梯度是Ｈｏ¨ｌｄｅｒ连续的条件仅仅是ｐ＞１．     

一类拟线性二阶微分方程解的振动与非振动的判定

本文讨论了拟线性微分方程（ｐ（ｘ＇））＇＋ｑ（ｔ）ｐ（ｘ）＝０,ｔ≥ｔ０,ｑ（ｔ）≥０（这里 ｐ（ｕ）＝｜ｕ｜ｐ－１ｕ,ｐ＞０是常数）的解的振动与非振动条件,并改进了文献［２］的结果．     

Fujita型反应扩散方程组整体解的存在性、非存在性与渐近性质

本文研究 Ｆｕｊｉｔａ型反应扩散方程组的初值问题：ｕｔ－△ｕ＝ａ１ｕ~α１－１ｕ＋ｂ１ｖ~β１－１ｖ,ｖｔ－△ｖ＝ａ２ｕ~α２－１ｕ＋ｂ２ｖ~β２－１ｖ,ｕ（Ｘ,０）＝ｕ０（Ｘ）,Ｖ（Ｘ,０）＝Ｖ０（Ｘ）,（Ｘ,ｔ）Ｒ~Ｎ ｘ Ｒ~＋,其中 ａｉ,ｂｉ≥ ０, αｉ,βｉ≥ １（ｉ＝ １,２）,给出了非负整体 Ｌ~ｐ解与古典解存在性与非存在性的一系列充分条件,并讨论了解的渐近性质．本文所用方法和所得结果与已有的工作［１－４］,有很大的不同,不但在某些方面推广了［１－５］,而且从某些方面改进了［１］的结果。     

双非线性抛物型方程非负解的一个性质

在Ｅ＾ｎ（０，∞）上讨论双非线性抛物型方程ａ／ａｔ（｜ｕ｜＾λ－２ｕ）－ｄｉｖ（｜△ｕ｜＾ｐ－２△ｕ）＝０在ｐ＞λ＞２的条件下，证明它的齐次Ｃａｕｃｈｙ问题非负整体解必是零解。     

EXISTENCE OF POSITIVE RADIAL SOLUTIONS FOR SOME QUASILINEAR ELLIPTIC EQUATIONSIN ANNULAR OMAINS

§１　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒｗｅｃｏｎｔｉｎｕｅｔｏｃｏｎｓｉｄｅｒｔｈｅｅｘｉｓｔｅｎｃｅｏｆｐｏｓｉｔｉｖｅｒａｄｉａｌｓｏｌｕｔｉｏｎｓｆｏｒｔｈｅｑｕａｓｉｌｉｎｅａｒｅｌｌｉｐｔｉｃｅｑｕａｔｉｏｎ－ｄｉｖ（｜Ｄｕ｜ｐ－２Ｄｕ）＝ｆ（ｕ）　ｉｎΩ,（１）ｕ（ｘ）＝０　ｏｎΩ,ｗｈｅｒｅｘ∈Ｒｎ,ｎ≥２,Ω＝｛ｘ：ａ＜｜ｘ｜＜ｂ,ａ,ｂ＞０｝,ａｎｄｐ＞１,ｆ∈Ｃ１（（０,∞））∩Ｃ０（［０,∞））ｓａｔｉｓｆｙｉｎｇｔｈｅｆｏｌｌｏｗｉｎｇｈｙｐｏｔｈｅｓｅｓ…     

包含测度的椭圆型方程解的Hlder连续性

考虑包含测度μ的椭圆型方程－ｄｉｖＡ（ｘ，ｕ，ｕ）＋Ｂ（ｘ，ｕ，ｕ）＝μ，在Ｇ内，ξ·Ａ（ｘ，ｕ，ξ）｜ξ｜ｐ－ｆ０（ｘ），１＜ｐ＜ｎ，｜Ａ（ｘ，ｕ，ξ）｜κ｜ξ｜ｐ－１＋ｆ１（ｘ），κ１，｜Ｂ（ｘ，ｕ，ξ）｜ｃ（ｘ）｜ξ｜γ＋ｆ２（ｘ），ｐ－１γｐ在γ＝ｐ－１的情况，为证有界解的Ｈｌｄｅｒ连续性，只需ｃ（ｘ）∈Ｌｎ（Ｇ）     

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.     

R2中非局部椭圆型方程基态解的存在性

本文考虑了一类非局部椭圆型方程-△u+V(x)u=(1/|x|μ*Q(x)F(u)/|x|β)Q(x)f(u)|x|β,x∈R^x,其中V是正的连续位势函数,0<μ<2,0≤β<1/2,2β+μ≤2,F(s)是f(s)的原函数.假设非线性项f(s)满足Trudinger-Moser型次临界指数增长,利用变分方法证明了该方程基态解的存在性.     

Solutions of Elliptic Equations ΔU+K(x)e2u=f(x)

In this paper we consider the elliptic equation Δu + K(x)e^{2u} = f(x), which arises from prescribed curvature problem in Riemannian geometry. It is proved that if K(x) is negative and continuous in R², then for any f ∈ L²_{loc} (R²) such that f(x) ≤ K(x), the equation possesses a positive solution. A uniqueness theorem is also given.     

A Multiplicity Result for a Singular and Nonhomogeneous Elliptic Problem in Rn

We establish sufficient conditions under which the quasilinear equation $$-div(|∇u|^{n-2}∇u)+V(x)|u|^{n-2}u=\frac{f(x,u)}{|x|^β}+εh(x) in \mathbb{R}^n,$$ has at least two nontrivial weak solutions in $W^{1,n} (\mathbb{R}^n)$ when ε > 0 is small enough, 0≤β < n, V is a continuous potential, f(x,u) behaves like $exp{γ|u|^{n/(n-1)}}$ as $|u|→∞$ for some γ > 0 and h≢ 0 belongs to the dual space of $W^{1,n} (\mathbb{R}^n)$.     

A sharp gradient estimate for the weighted p-Laplacian

Let M be an n-dimensional complete noncompact Riemannian manifold with sectional curvature bounded from below, d?? = e ^h (x) dV (x) the weighted measure and ??_??,p the weighted p-Laplacian. In this paper we consider the non-linear elliptic equation $$ \Delta _{\mu ,p} u = - \lambda _{\mu ,p} |u|^{p - 2} u $$ for p ?? (1, 2). We derive a sharp gradient estimate for positive smooth solutions of this equation. As applications, we get a Harnack inequality and a Liouville type theorem..     

Geometry of unbounded domains,poincaré inequalities and stability in semilinear parabolic equations

We investigate thc close relations existing between certain geometric properties of domains Ω of R^N, the validity of Poincark inequalities in Ω, and the behavior of solutions of semilinear parabolic equations. For the equation u_t-△u=|u|^p-1 we obtain a purely geometric, necessary and sufficient condition on Ω, for the 0 solution to be asymptotically (and exponentially) stable in L^r(ω)1<r<∞ when r is supercritical(r>N(p-1)/2 . The condition is that the inradius of Ω be finite. The result is different for r critical. For the equation u_t-△u=u^p-μ|u|^q,q≥p>1,μ>0 we prove that the finiteness of the inradius is a necessary and sufficient condition for global existence and boundedness of all nonnegative solutions.     

Existence of Nontrivial Weak Solutions to Quasi-linear Elliptic Equations with Exponential Growth

In this paper, we study the existence of nontrivial weak solutions to the following quasi-linear elliptic equations $$-Δ_nu+V(x)|u|^{n-2}u=\frac{f(x,u)}{|x|^β}, x ∈ R^n(n ≥ 2),$$ where $-Δ_nu=-div(|∇u|^{n-2}∇u), 0 ≤β < n, V:R^n→R$ is a continuous function, f (x,u) is continuous in $R^n×R$ and behaves like $e^{αu^{\frac{n}{n-1}}}$ as $u→+∞$.     

Existence Results for Superlinear Elliptic Equations with Nonlinear Boundary Value Conditions

In this paper, we study the existence of solutions for the following superlinear elliptic equation with nonlinear boundary value condition $$\left\{ {\begin{array}{*{20}{c}} { - \Delta u + u = {{\left| u \right|}^{r - 2}}u}&{in\;\Omega ,\;\;} \\ {\frac{{\partial u}}{{\partial v}} = {{\left| u \right|}^{q - 2}}u}&{on\;\partial \Omega ,} \end{array}} \right.$$ where Ω ⊂ ℝN, N ≥ 3 is a bounded domain with smooth boundary. We will prove the existence results for the above equation under four different cases: (i) Both q and r are subcritical; (ii) r is critical and q is subcritical; (iii) r is subcritical and q is critical; (iv) Both q and r are critical.     

带有阻尼项的偏泛函微分方程解的振动性

本文研究带有阻尼项的双曲型时滞偏微分方程 2 t2 u(x,t) +m(t) u t=a(t)△ u(x,t) +b(t)△ u(x,ρ(t) ) -q(t) f (u(x,σ(t) ) ,(x,t)∈ G≡Ω× R+ (1 )其中 ,R+=[0 ,+∞ ) ,Ω是一个具有逐段光滑边界的有界区域 .利用平均法和微分不等式方法得到方程 (1 )的若干新的振动准则 .     

Ground States for Singularly Perturbed Planar Choquard Equation with Critical Exponential Growth

In this paper, we are dedicated to studying the following singularly Choquard equation $$ -\varepsilon^2\Delta u+V(x)u=\varepsilon^{-\alpha}\left[I_{\alpha}\ast F(u)\right]f(u),\ \ \ \ x\in\R^2,$$ where $V(x)$ is a continuous real function on $\R^2$, $I_{\alpha}:\R^2\rightarrow\R$ is the Riesz potential, and $F$ is the primitive function of nonlinearity $f$ which has critical exponential growth. Using the Trudinger-Moser inequality and some delicate estimates, we show that the above problem admits at least one semiclassical ground state solution, for $\varepsilon>0$ small provided that $V(x)$ is periodic in $x$ or asymptotically linear as $|x|\rightarrow \infty$. In particular, a precise and fine lower bound of $\frac{f(t)}{e^{\beta_{0} t^{2}}}$ near infinity is introduced in this paper.