LARGE TIME BEHAVIOR OF SOLUTIONS TO NEWTONIAN FILTRATION EQUATIONS WITH SOURCES

This article is concerned with large time behavior of solutions to the Neumann or Dirichlet problem for a class of Newtonian filtration equations |x|λ+k ■u■ t = div(|x|k▽um) + |x|λ+kupwith 0 m 1, p 1, λ≥ 0, k ∈ R. An interesting phenomenon is that there exist two thresholds k∞ and k1 for the exponent k, such that the critical Fujita exponent pc for p exists and is finite if k ∈ (k∞, k1), otherwise, pc is infinite or does not exist.     

THE REGULARITY OF QUASI-MINIMA AND ω-MINIMA OF INTEGRAL FUNCTIONALS

In this article, we have two parts. In the first part, we are concerned with the locally Hlder continuity of quasi-minima of the following integral functional ∫Ωf(x, u, Du)dx, (1) where Ω is an open subset of Euclidean N-space (N ≥ 3), u:Ω→ R,the Carath′eodory function f satisfies the critical Sobolev exponent growth condition |Du|p* |u|p*-a(x) ≤ f(x,u,Du) ≤ L(|Du|p+|u|p* + a(x)), (2) where L≥1, 1pN,p* = Np/N-p , and a(x) is a nonnegative function that lies in a suitable Lp space. In the second part, we study the locally Hlder continuity of ω-minima of (1). Our method is to compare the ω-minima of (1) with the minima of corresponding function determined by its critical Sobolev exponent growth condition. Finally, we obtain the regularity by Ekeland’s variational principal.     

CRITICAL EXPONENTS AND CRITICAL DIMENSIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH SINGULAR COEFFICIENTS

Let B1 ■ RNbe a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical Sobolev exponent and singular coefficients:-div(|▽u|p-2▽u) = |x|s|u|p*(s)-2u + λ|x|t|u|p-2u, x ∈ B1,u|■B1= 0,where t, s -p, 2 ≤ p N, p*(s) =(N+s)p N-pand λ is a real parameter. We show particularly that the above problem exists infinitely many radial solutions if the space dimension N p(p- 1)t + p(p2- p + 1) and λ∈(0, λ1,t), where λ1,t is the first eigenvalue of-△p with the Dirichlet boundary condition. Meanwhile, the nonexistence of sign-changing radial solutions is proved if the space dimension N ≤(ps+p) min{1,p+t p+s}+p2p-(p-1) min{1,p+t p+s}and λ 0 is small.     

EXISTENCE OF INFINITELY MANY SOLUTIONS FOR ELLIPTIC PROBLEMS WITH CRITICAL EXPONENT

This paper is concerned with the following nonlinear Dirichlet problem:{-Δpu=|u|^p*-2 u λf(x,u) x∈Ω；u=0 x∈эΩ} whereΔp^u = div(|∧u|^p-2∧u) is the p-Laplacian of u,Ω is a bounded in R^n(n≥3），1＜p＜n, p=pn/n-p is the critical exponent for the Sobolev imbedding,λ＞0 and f(x,u)satisfies some conditions. It reaches the conclusion that this problem has infinitely many solutions. Some results as p=2 or f(x,u) = |u|^q-2 u, where 1＜q＜p, are generalized.     

Multiple solutions for weighted nonlinear elliptic system involving critical exponents

In this paper,by using the idea of category,we investigate how the shape of the graph of h(x)affects the number of positive solutions to the following weighted nonlinear elliptic system:-div(|x|-2au)-μu|x|2(a+1)=αα+βh(x)|u|α-2|v|βu|x|b2*(a,b)+λK1(x)|u|q-2u,in,-div(|x|-2av)-μv|x|2(a+1)=βα+βh(x)|u|α|v|β-2v|x|b2*(a,b)+σK2(x)|v|q-2v,in,u=v=0,on,where 0∈is a smooth bounded domain in RN(N 3),λ,σ0 are parameters,0μμa(N-2-2a2)2;h(x),K1(x)and K2(x)are positive continuous functions in,1 q2,α,β1 andα+β=2*(a,b)(2*(a,b)2N N-2(1+a-b),is critical Sobolev-Hardy exponent).We prove that the system has at least k nontrivial nonnegative solutions when the pair of the parameters(λ,σ)belongs to a certain subset of R2.     

关于含临界指数的p-平均曲率算子

This paper is concerned with the existence of positive solutions of the following Dirichlet problem for p-mean curvature operator with critical exponent: -div((1 |▽u|~2)(p-2)/2▽u)=λu~(p*-1) μu~(q-1),u＞0,x∈Ω, u=0,x∈■Ω, where u∈W_0~(1,p)(Ω),Ωis a bounded domain in R~N(N＞p＞1)with smooth boundary ■Ω,2＜=p＜=q＜=P~*,P~*=(Np)/(N-p),λ,P＞0.It reaches the conclusions that this problem has at least one positive solution in the different cases.It is discussed the existences of positive solutions of the Dirichlet problem for the p-mean curvature operator with critical exponent by using Nehari-type duality property firstly.As p=2,q=p,the result is correspond to that of Laplace operator.     

Multi-peak nodal solutions for a two-dimensional elliptic problem with large exponent in weighted nonlinearity

We consider the boundary value problem u +p|x2α||u|-1u = 0,-1 < α = 0, in the unit ball B with the homogeneous Dirichlet boundary condition, when p is a large exponent. By a constructive way, we prove that for any positive integer m, there exists a multi-peak nodal solution up whose maxima and minima are2 located alternately near the origin and the other m points qll=(λ cosπ(l-1)2, λ-1)msinπ(m), l = 2,, m + 1,such that as p goes to +∞,m+1pα|x2||upp|-1up 8πe(1 + α)δ0 + 8πe(-1)l-1δl=2re λ∈(0, 1), m is an odd number with(1 + α)(m + 2)-1 > 0, or m is an even,ql whe number. The same techniques lead also to a more general result on general domains.     

On Critical Fujita Exponents for a Degenerate Parabolic Equation With

Our interest is to determine the critical Fujita exponent concerned withthe following initial-boundary value problem ut= Δum, x ∈RN+, t>0, (1) u(x,0)=u0(x), x∈RN+, (2) -(um)/(x1)=up, x1=0, t>0, (3) where RN+=(x1, x′)| x′∈R{N-1, x1>0,m>1, p>0, and u0(x) is a nonnegative bounded function with compact supportsatisfying thecompatibility condition -(um0(x))/(x1)=up0(x), x1=0. We call p0 the critical global existence exponent if ithas the following property: if p>p0, there always exist nonglobalsolutions of the problem (1)--(3) while if 0pc small data solutionsexist globally in time while large data solutions are nonglobal.     

Existence and multiplicity of normalized solutions for a class of fractional Choquard equations

In this paper, we study the existence and multiplicity of solutions with a prescribed L2-norm for a class of nonlinear fractional Choquard equations in RN:(-△)su-λu =(κα*|u|p)|u|p-2u,where N≥3,s∈(0,1),α∈(0,N),p∈(max{1 +(α+2s)/N,2},(N+α)/(N-2s)) and κα(x)=|x|α-N. To get such solutions,we look for critical points of the energy functional I(u) =1/2∫RN|(-△)s/2u|2-1/(2p)∫RN(κα*|u|p)|u|p on the constraints S(c)={u∈Hs(RN):‖u‖L2(RN)2=c},c >0.For the value p∈(max{1+(α+2s)/N,2},(N+α)/(N-2s)) considered, the functional I is unbounded from below on S(c). By using the constrained minimization method on a suitable submanifold of S(c), we prove that for any c>0, I has a critical point on S(c) with the least energy among all critical points of I restricted on S(c). After that,we describe a limiting behavior of the constrained critical point as c vanishes and tends to infinity. Moreover,by using a minimax procedure, we prove that for any c>0, there are infinitely many radial critical points of I restricted on S(c).     

EXISTENCE AND NONEXISTENCE OF GLOBAL POSITIVE SOLUTIONS FOR DEGENERATE PARABOLIC EQUATIONS IN EXTERIOR DOMAINS

This article deals with the degenerate parabolic equations in exterior domains and with inhomogeneous Dirichlet boundary conditions. We obtain that pc = (σ+m)n/(n-σ-2) is its critical exponent provided max{-1, [(1-m)n-2]/(n+1)} σ n-2. This critical exponent is not the same as that for the corresponding equations with the boundary value 0, but is more closely tied to the critical exponent of the elliptic type degenerate equations. Furthermore, we demonstrate that if max{1, σ + m} p ≤ pc, then every positive solution of the equations blows up in finite time; whereas for ppc, the equations admit global positive solutions for some boundary values and initial data. Meantime, we also demonstrate that its positive solutions blow up in finite time provided n ≤σ+2.     

${\mbox{\boldmath $R$}}^N$上奇异非线性多调和方程的正整体解

本文研究形如△((△nu)(p-1) )=f(|x|,u,|(?)u|)u-β,x∈RN的奇异非线性多调和方程在RN上的正整体解,此处P>1,β≥0是常数,n是自然数,f:R × R ×R →R 是一个连续函数, ξδ*:=sign(ξ)·|ξ|δ,,ξ∈R,δ>0,给出了该类方程具有无穷多个其渐进阶刚好为|x|2n的正整体解的充分条件与必要条件.这些结论可以推广到更一般的方程.     

奇异非线性$p-$调和方程的一类正整体解

设p>1,β≥0是常数, n是自然数, 是一个连续函数.本文研究形如的奇异非线性p-调和方程的正整体解,给出了该类方程具有无穷多个其渐近阶刚好为|x|(2n-2)(当|x|→∞时)的径向对称的正整体解的若干充分条件.     

自正则化和Davis大数律和重对数律的精确渐近性

本文证明了自正则化Davis大数律和重对数律的精确渐近性, 即 {\heiti\bf 定理1}\hy 设$\ep X=0$, 且$\ep X^2I_{(|X|\leq x)}$在无穷远处是缓变函数, 则$\lim_{\varepsilon\searrow0}\varepsilon^2\tsm_{n\geq3}\frac{1}{n\log n}\pr\Big(\Big|\frac{S_n}{V_n}\Big|\geq\varepsilon\sqrt{\log\log n}\Big)=1.${\heiti\bf 定理2}\hy 设$\ep X=0$, 且$\ep X^2I_{(|X|\leq x)}$在无穷远处是缓变函数, 则对本文证明了目正则化Davis大数律和重对数律的精确渐近性,即定理1设EX=0,且EX~2I_(|x|≤x)在无穷远处是缓变函数,则■定理2设EX=0,且EX~2I_(|x|≤x)在无穷远处是缓变函数,则对0≤δ≤1,有■其中N为标准正态随机变量．     

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.     

Multibump bound states for quasilinear Schrödinger systems with critical frequency

In this paper, we are concerned with the multibump solutions for the following quasilinear Schrödinger system in ${\mathbb{R}^N}$ : $$\left\{\begin{array}{ll}-\Delta{u} + \lambda{a(x)u} - \frac{1}{2}(\Delta|u|^2)u = \frac{2\alpha}{\alpha + \beta}|u|^{\alpha-2}|\upsilon|^\beta u, \\-\Delta{\upsilon} + \lambda{b(x)\upsilon} - \frac{1}{2}(\Delta|\upsilon|^2)\upsilon = \frac{2\beta}{\alpha + \beta}|u|^\alpha|\upsilon|^{\beta-2} \upsilon, \\u(x) \rightarrow 0, \upsilon(x) \rightarrow 0 \quad as|x| \rightarrow \infty,\end{array}\right.$$ where λ > 0 is a parameter, α, β > 2 satisfying α + β < 2 · 2*, here ${2^{*} = \frac{2N}{N-2}}$ is the critical Sobolev exponent for ${N \geq 3}$ and a(x), b(x) are nonnegative potentials. Using variational methods, we prove that if the zero sets of a(x) and b(x) have several common isolated connected components ${\Omega_{1}, . . . ,\Omega_{k}}$ such that the interior of ${\Omega_{i} (i = 1, 2, . . . , k)}$ is not empty and ${\partial\Omega_{i} (i = 1, 2, . . . , k)}$ is smooth, then for λ sufficiently large, the system admits, for any nonempty subset ${J \subset \{1, 2, . . . , k\}}$ , a solution which is trapped in a neighborhood of ${\cup_{j\epsilon{J}} \Omega_{j}}$ .     

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 $|x|→∞.$     

Local Continuity and Harnack’s Inequality for Double-Phase Parabolic Equations

Potential Analysis - We consider parabolic equations of the form $$ u_{t}-\text{div} \left( |\nabla u|^{p-2}\nabla u+ a(x,t)|\nabla u|^{q-2}\nabla u\right)= 0, a(x,t)\geq 0. $$ In the range $\frac...     

On ground state of fractional $p$-Kirchhoff equation involving subcritical and critical exponential growth

In this paper, we concern the existence of nontrivial ground state solutions of fractional $p$-Kirchhoff equation $$\left\{\begin{array}{ll} m\left(\|u\|^p\right) [(-\Delta)_p^su+V(x)|u|^{p-2}u] =f(x,u) \quad\text{in}\, \mathbb{R}^N, \vspace{0.2 cm}\\ \|u\|=\left(\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy +\int_{\mathbb{R}^N}V(x)|u|^pdx\right)^{\frac{1}{p}}, \end{array}\right.$$ where $m:[0,+\infty)\rightarrow [0,+\infty)$ is a continuous function, $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator with $0相似文献   

一类具有奇异灵敏度的logistic趋化系统解的渐近行为

本文在无边界流的光滑有界区域$\Omega\subset\mathbb{R}^n~(n>2)$上研究了具有奇异灵敏度及logistic源的抛物-椭圆趋化系统$$\left\{\begin{array}{ll}u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+r u-\mu u^k,&x\in\Omega,\,t>0,\\ 0=\Delta v-v+u,&x\in\Omega,\,t>0\end{array}\right.$$ 其中$\chi$, $r$, $\mu>0$, $k\geq2$. 证明了若当$r$适当大, 则当$t\rightarrow\infty$时该趋化系统全局有界解呈指数收敛于$((\frac{r}{\mu})^{\frac{1}{k-1}}, (\frac{r}{\mu})^{\frac{1}{k-1}})$.     

Global integrability for minimizers of anisotropic functionals

We consider integral functionals in which the density has growth p _i with respect to ${\frac{\partial u}{\partial x_i}}$ , like in $$\int\limits_{\Omega}\left( \left| \frac{\partial u}{\partial x_1}(x) \right|^{p_1} + \left|\frac{\partial u}{\partial x_2}(x)\right|^{p_2} + \cdots + \left|\frac{\partial u}{\partial x_n}(x) \right|^{p_n} \right) dx.$$ We show that higher integrability of the boundary datum forces minimizer to be more integrable.