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).     

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.     

On the Greatest Common Di visor of u − 1 and v − 1 with u and v Near ${\cal S}$- units

In [2], it was shown that if a and b are multiplicatively independent integers and ɛ > 0, then the inequality gcd (aⁿ − 1,bⁿ − 1) < exp(ɛn) holds for all but finitely many positive integers n. Here, we generalize the above result. In particular, we show that if f(x),f₁(x),g(x),g₁(x) are non-zero polynomials with integer coefficients, then for every ɛ > 0, the inequality holds for all but finitely many positive integers n.     

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...     

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.     

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。     

An efficient algorithm for generating all k-subsets (1 ⩽ k ⩽ m ⩽ n) of the set [1, 2,…, n] in lexicographical order

We consider the problem of generating all k-subsets (1 ? k ? m ? n) of the set [1, 2, …, n] in lexicographical order. The running time per k-subset is shown to be constant. The experimental data suggest that our algorithm is about 25% faster than the algorithm proposed by A. Nijenhuis and H. S. Wilf (“Combinatorial Algorithms,” 2nd ed., Academic Press, New York, 1978).     

Remarks on bifurcations ofu + μu − u k=0 (4≤kεZ +)

In this paper, we investigate the bifurcations of one class of steady-state reaction-diffusion equations of the formu″ + μu − u ^k=0, subjectu(0)=u(π)=0, where μ is a parameter, 4≤kεZ ⁺. Using the singularity theory based on the Liapunov-Schmidt reduction, some satisfactory results are obtained. This work is supported by the National Natural Science Foundation of China (No.19971057) and the Youth Science Foundation of Shanghai Municipal Commission of Education (No.99QA66).