图上p-Laplace算子的特征值估计

设G(V,E)为满足■(m,K)条件的连通有限图,其中p≥2,m 0,K≤0.本文在G上考虑了方程?pu=-λ_p|u|~(p-2)u解的椭圆型梯度估计,其中?_p为p-Laplace算子.作为应用,导出了G上?_p的第一非零特征值的下界估计.     

含临界指数的拟线性椭圆系统正对称解的存在性（英文）

本文讨论一类拟线性椭圆型系统-Δpu=μ|u|p-2 u|x|p+2αQ(x)(α+β)|x|s|u|α-2 u|v|β+σ1|u|q1-2 u,x∈Ω,-Δpv=μ|v|p-2v|x|p+2βQ(x)(α+β)|x|s|u|α|v|β-2v+σ2|v|q2-2v,x∈Ω,u=v=0,x∈Ω,其中Δpu=div(|▽u|p-2▽u)是p-Laplacian,2≤pN,ΩRN是一个有界光滑区域,0∈Ω,且Ω关于O(N)的一个闭子群G对称,0≤μ,=((N-p)/p)p,σ1,σ2≥0,0≤sp,α,β1满足α+β=p*(s)=(N-s)p/(N-p),pq1,q2p*=Np/(N-p),Q(x)是Ω上的连续G对称函数.应用Palais对称临界原理和变分方法,我们建立了该系统几个全新的正G-对称解的存在性结果.     

ON THE SINGULAR VARIATIONAL PROBLEMS

The authors deal with the singular variational problem S(a,b,λ0):=infu∈E,u(≡／)0 ∫RN(||X|-a(△)u|m ∫|x|-(a 1)m|u|m)dx/(∫RN||X|-bU|P dx)m/p as well as (S)=(S)(a,b,λ1,λ2):=u,ν,E∈,u(u,ν)(≡／)(1,1) ∫RN J(u,ν)dx/(∫RN|x|-bp|u|α|ν|βdx)m/p, whereJ(u, v) = ||x|- au|m λ1|x|- (a 1)m|u|m ||x|- av|m λ2|x|- (a 1)m|v|m,N ≥ m 1 ＞ 2, 0 ≤ a ＜ N-m/m, a ≤ b ＜ a 1 and p = p(a,b) = α β =Nm/N-m m(b-a), α, β≥ 1, E = D1,mα(RN). The aim of this paper is to show the existence of minimizer for S(a, b, A0) and S(a, b, λ1, λ2).     

R~N上临界增长的椭圆方程无穷多解的存在性

本文证明了RN上的拟线性椭圆型方程-div（|Du|p-2Du）+|u|p-2u=λ（x）·|u|α-2u+a（x）|u|s-2u+b（x）|u|p＊-2u在W1,p（RN）中无穷多解的存在性,其中N≥3,2≤p相似文献   

具位势项的快扩散方程的第二临界指标

This paper considers a fast diffusion equation with potential ut= um V (x)um+upin Rn×(0,T), where 1 2αm+n< m ≤ 1, p > 1, n ≥ 2, V (x) ～ω|x|2with ω≥ 0 as |x| →∞,and α is the positive root of αm(αm + n 2) ω = 0. The critical Fujita exponent was determined as pc= m +2αm+nin a previous paper of the authors. In the present paper,we establish the second critical exponent to identify the global and non-global solutions in their co-existence parameter region p > pcvia the critical decay rates of the initial data.With u0(x) ～ |x| aas |x| →∞, it is shown that the second critical exponent a =2p m,independent of the potential parameter ω, is quite different from the situation for the critical exponent pc.     

一类带Sobolev-Hardy指数的椭圆型方程组无穷多球对称解的存在性

考虑了一类带Sobolev-Hardy指数的椭圆型方程组{-Δu-μu/|x|2=α/α+β|μ|α-2u|v|β/|x|s+σp/p+q|u|p-2u|v|q,x∈B,-Δu-μu/|x|2=β/α+β|μ|α|v|β-2v/|x|s+σp/p+q|u|p|v|q-2,x∈B,其中0≤μμ,-4,μ=((N-2)~2)/4,σ0,0≤s2,N6+s,α+β=2~*(s)=(2(N-s))/(N-2),p,q≥1,2≤p+q2~*(s),B■R~N为以原点为心的一个开球.利用逼近方法及喷泉定理,得到了上述方程组无穷多个球对称解的存在性.     

类渐近线性p＆q—Laplace方程弱解的全局衰减性

该文研究椭圆型方程{-Δpu+m|u|p-2u-Δqu+n|u|q-2u=g(x,u),x∈RN,u∈ W1,p(RN)∩W1,q(RN)弱解在全空间RN上的衰减性,其中m,n≥0,N≥3,1相似文献   

Liouville theorem for Choquard equation with finite Morse indices

In this article, we study the nonexistence of solution with finite Morse index for the following Choquard type equation-△u=∫RN|u(y)|p|x-y|αdy|u(x)|p-2u(x) in RN where N ≥ 3, 0 α min{4, N}. Suppose that 2 p (2 N-α)/(N-2),we will show that this problem does not possess nontrivial solution with finite Morse index. While for p=(2 N-α)/(N-2),if i(u) ∞, then we have ∫_RN∫_RN|u(x)p(u)(y)~p/|x-y|~α dxdy ∞ and ∫_RN|▽u|~2 dx=∫_RN∫_RN|u(x)p(u)(y)~p/|x-y|~αdxdy.     

Morse指数与带权的椭圆方程的Liouville型定理

该文研究全空间R~N上带权的半线性椭圆型方程-△u=|x|~α|u|~(p-1)u,x∈R~N与半空间R_+~N={x∈R~N:x_N0}上带权的半线性椭圆型问题-△u=|x|~α|u|~(p-1)u,x∈R_+~N,u|?R_+~N=0的Liouville型定理,其中N≥3,α-2.证明了,当1p(N+2α+2)/(N-2)时,上述问题的Morse指数有限的有界解只能是零解.     

广义拟线性Schr?dinger方程基态解的存在性

本文旨在研究如下的广义拟线性Schr?dinger方程-div(g~2(u)▽u)+g(u)g′(u)|▽u|~2+V (x)u=h(u), x∈R~N,其中N≥3, g:R→R~+是一个可微的偶函数且存在α≥1使得lim~(t→+∞)g(t)/t~(α-1)=β 0; h:R→[0,+∞)是一个非线性函数且包含情形:h(t)=|t|~(p-2)t (2 p α2*);位势函数V (x):R~N→R为正.结合变量替换和变分技巧,本文证明了上述问题存在一个正的基态解.     

带有非紧条件的拟线性Schrodinger-Poisson系统非平凡解的存在性

本文研究了如下带有非紧条件的拟线性Schrodinger-Poisson系统{-△u+V(x)u+Фu+k/2u△u2=λ|u|^p-2u+f(u),x ∈R^3,-ΔФ=u^2,x∈R^3, 其中κ<0,λ>0,p≥12,f∈C(R,R),V∈C(R3,R).文中首先构造截断函数,利用集中紧性原理和逼近的方法,得到了截断后系统非平凡解的存在性;然后利用Moser迭代技巧,讨论上述系统非平凡解的存在性.     

Linking solutions for quasilinear equations at critical growth involving the “1-Laplace” operator

We show that the problem at critical growth, involving the 1-Laplace operator and obtained by relaxation of \({-\Delta_1 u=\lambda |u|^{-1}u+|u|^{1^*-2} u}\) , admits a nontrivial solution \({u \in BV(\Omega)}\) for any λ ≥ λ₁. Nonstandard linking structures, for the associated functional, are recognized.     

含Hardy位势的拟线性椭圆型方程非平凡解的存在性

设0∈Ω∈R^N,(N≥2)为有界光滑区域,利用山路定理,考虑如下一类含Hardy位势的拟线性椭圆型方程非平凡解的存在性:-△u-u△(|u|N,(N≥2)为有界光滑区域,利用山路定理,考虑如下一类含Hardy位势的拟线性椭圆型方程非平凡解的存在性:-△u-u△(|u|²)=μu/|x|2)=μu/|x|²+λg(x,u),x∈Ω,其中μ>0,λ>0为常数,g(x,u)为Caratheodory函数.     

本刊英文版Vol.37(2021),No.11论文摘要（英文）

<正>p-Laplacian Equations on Locally Finite Graphs Xiao Li HAN Meng Qiu SHAO Abstract This paper is mainly concerned with the following nonlinear p-Laplacian equation-Δ_pu(x)+(λa(x)+1)|u|~(p-2)(x)u(x)=f(x,u(x)),in V on a locally finite graph G=(V,E) with more general nonlinear term,whereΔ_p is the discrete p-Laplacian on graphs,p≥2.Under some suitable conditions on f and a(x),we can prove that the equation admits a positive solution by the Mountain Pass theorem and a ground state solution u_λvia the method of Nehari manifold,for anyλ> 1.In addition,asλ→+∞,we prove that the solution u_λconverge to a solution of the following Dirichlet problem     

一类临界增长非线性椭圆方程的解

利用非光滑临界点理论 ,本文证明了一类临界增长非线性椭圆方程-div(A(x ,u) ｜ u｜p-2 u) 1pA′u(x ,u) ｜ u｜p =g(x ,u) ｜u｜p -2 u ,u=0 ,　 Ω ; Ω 非平凡正解的存在性 .其中 1 相似文献   

A-调和方程弱解的双权Caccioppoli型不等式

研究形如div A(x,u(x))=0的A-调和方程,证明了其弱解满足局部Aλr双权Caccioppoli型不等式.其中算子A:Ω×Rn→Rn满足如下条件:对于正常数0   

Regular solutions of initial-boundary value problems for linear and nonlinear wave-equations I

This paper deals with the question of the existence of classical solutions for the equations $$\frac{{\partial ^{2} u}{\partial t^{2} }} + \sum_{\begin{subarray}{l} |\alpha| \leqslant m \\ | \beta | \leqslant m \end{subarray}} D^{\alpha} (A_{\alpha \beta } (x,t) D^{\beta} u) = f (t,x,u)$$ on [0,T] × G. G is a bounded or unbounded domain; the differential operator in the space variables is elliptic; the initial values of u are prescribed and D^αu (t,x) vanishes for (t,x) ∈ [0,T] × ?G, |α|≤ m?1. First we develop a method for solving regularly linear wave equations. In contrast to the usual compatibility conditions, our method requires less differentiability in t but imposes some boundary conditions on f(t). It allows some applications to nonlinear problems which will be treated in the second part of this paper and which e.g. enable us to solve ?² u/?t²?A(t)u+u³=f.     

Scattering for Semilinear Wave Equation with Small Data in High Space Dimensions

In this paper we study the scattering theory for the semilincar wave equation u_{tt} - Δu = F(u(t, x), Du(t, x)) in R^n (n ≥ 4) with smooth and small data. We show that the scattering operator exists for the nonlinear term F = F(λ) = O(|λ|^{1, α}), where α is an integer and satisfies α ≥ 2, n = 4; α ≥ I, n ≥ 5.     

一类奇异p-Laplace方程无穷多解的存在性

讨论了一类具有奇异系数的p-Laplace问题-Δpu-μ|u|u|x|p=u|x|tu+λuq-2u,x∈Ω,u=0,x∈Ω无穷多解的存在性,其中N≥3,Ω是RN中一有界光滑区域,0∈Ω,Δpu=-div(|▽u|p-2▽u),0≤μ<μ=(N-p)ppp,10,1相似文献