方程u″-△u+|u|~pulog~k(1+|u|)=f的存在唯一性

我们知道非线性双曲方程是在相对论量子力学的研究中提出的问题(参看[1][2])。在[3][4][5]中,人们研究了这个方程在Sobolev空间中的可解性,本文用Sobolev-Orlicz空间理论和Faedo-galerkin方法,讨论非线性双曲方程的初边值问题的可解性。在(P,K)平面上,我们给出了这个方程的存在区域和唯一区域,并指出已有结果包括在K=0的特殊情形中。     

On radially symmetric solutions of the equation ?Δu + u = |u| p-1 u. An ODE approach

Questions of the existence in a ball of radially symmetric solutions of the equation indicated in the title with the Dirichlet zero boundary conditions are studied in many publications and generally speaking, there was obtained more or less complete answer on these questions. It is known now that if the dimension of the space d????3 and 1 <?p?<?(d?+ 2)/(d ? 2) or if d?=?2 and p?>?1, then for any integer l??? 0 this problem in a ball or in the entire space ${x \in \mathbb {R}^d}$ has a radially symmetric solution with precisely l zeros as a function of r?=?|x|. If d??? 3 and p????(d?+?2)/(d ? 2), then the problem in the entire space has no nontrivial solution. For the first time, this problem was studied by a variant of the variational method. However, it is known to the specialists in the field that it is also interesting to obtain the same results by using methods of the qualitative theory of ODEs. In the present article, we shall give a simple proof of the result above in this way. An earlier proof of this result of the other authors is essentially more complicated than our one.     

方程△u+|u|~(p-1)u+λu=0的径向解的一些注记

当n≥7时,已经证明对任意的λ∈(0,λ_1)以及任意整数k≥0,R~n中的单位球B(0,1)上的方程⊿u+|u|~(p-1)u+λu=0在H_0~1(B(0,1))中必有一个径向解具k个结点。本文证明当3≤n≤6时这一结果不再成立。还讨论了上述方程径向正解的唯一性。     

Liouville-type results for solutions of −Δu=|u|p−1u on unbounded domains of RN

In this Note we study solutions, possibly unbounded and sign-changing, of the equation $?\mathrm{\Delta }u={|u|}^{p?1}u$ on unbounded domains of ${\mathbb{R}}^N$ with $N?2$ and $p>1$. We prove some Liouville-type results and a classification theorem for $C^2$ solutions belonging to one of the following classes: stable solutions, finite Morse index solutions and solutions which are stable outside a compact set. We also extend, to smooth coercive epigraphs, the well-known results of Gidas and Spruck concerning non-negative solutions of the considered equation. To cite this article: A. Farina, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

偏微分方程△^2u—s△u＋k^2u=0边值问题的MRM边界变分方程

导出边值问题△^2u-s△u k^2u=0；x∈Ω∪Ω‘∪→R^2;u/Γ=u0;δu/δn/Γ=g0的定解问题，MRM边界变分方程，全平面解的表达式，从中可以以看出，MRM边界变分方程中只包含弱奇异积分核，并且自动消除了原第一，二MRM边界积分方程中出现的强奇异积分核，问题解的表达式后并不加任何多项式，因而也不需要引入Lagrange乘子求解该项，这给边界元数值求解过程带来极大的方便，数值分析结果表明该方法具有明显优势。     

Banach空间中—u‘’∈F（t,u（t）,u（t））的解

本文给出了弱序列完备Ｂａｎａｃｈ空间中二阶微分包含的解的存在性定理，推广了「１」中的结果到集值情形，同时由于利用已有的定理「２」，简化了类似「１」宫的证明。     

A note on the problem −Δu=λu+u|u|2*−2

The dual approach to the problem –u=u+u|u|^2*–2, uþe ₀ ¹ (gW), permits a simple proof of a recent existence result [5] and allows extensions of this result to similar problems also with asymmetric mnonlinearities.Supported by min P.I., Gruppo Naz. (40%) Calcolo delle Variazioni...     

On the equation −Δu + e u − 1 = 0 with measures as boundary data

If Ω is a bounded domain in ${\mathbb{R}^N}$ , we study conditions on a Radon measure μ on ?Ω for solving the equation ?Δu + e ^u ? 1 = 0 in Ω with u = μ on ?Ω. The conditions are expressed in terms of Orlicz capacities.     

Δu=e u und die automorphen Funktionen

Ohne Zusammenfassung     

方程—Δu=g(x,u,Du)正解的先验估计和存在性

本文研究了一类超线性方程Dirchlet问题在Q中,正解的先验估计和存在性。其中函数g(x,t,P)关于变元t∈R 和p∈R~N都可以按幂函数的形式真t~α和|p|~s(α>1,β≥0)的形式增长。本文改进了文献[1]的结果。     

方程~2u/x~1x~1 ~2u/x~2x~2=f(u)的Bcklund变换

本文利用Wahlquist-Estabrook过程(WEP)研究了方程(?)~2u/(?)x~1(?)x~1 (?)~2u/(?)x~2(?)x~2=f(u)(这里f是任意函数)的B(?)cklund变换.我们发现该方程存在B(?)cklund变换的充分条件是d~2f/du~2=λf.我们所得到的结果的一个特殊情况就是Leibbrandt的结论.     

The integrable nonlinear degenerate diffusion equationu t=[f(u)u x −1 x and its relatives

The nonlinear diffusion equationu _t=[f(u)g(u _x )] arises in recent models of turbulent transport and of stress dissipation in rock blasting. A Lie point symmetry analysis produces many similarity reductions of exponential and power-law forms, and reveals that for all choices off the equation is always integrable wheng(u _x )=1/u _x . We identify the functionsf(u) which guarantee equivalence to the linear heat equation. For all other choices off, the linear canonical form leads to a self-adjoint differential equation by separation of variablesx andt. We construct a number of explicit solutions with simple boundary conditions, which illustrate behavior in the vicinity of the degenerate region withu _x =. If zero flux and constant concentration are maintained on free boundaries, then steep concentration gradients may evolve from smooth initial conditions. For other boundary conditions, unlike the examples of strong degeneracy, smoothing will occur at initial step discontinuities.     

Ganze Lösungen der Differentialgleichung Δu=e u

Ohne Zusammenfassung