A minimum principle for the problem of St-Venant in {\mathbb{R}^N, N \geq 2}

This paper deals mainly with the St-Venant problem in a convex domain ?? of ${\mathbb{R}^N, N \geq 2}$ . A minimum principle for a combination of the stress function ${\psi}$ and ${|\nabla \psi|}$ is derived. Some possible applications are indicated.     

Multibubble analysis on N-Laplace equation in {\mathbb{R}^N}

In this note, we describe the asymptotic behavior of sequences of solutions to N-Laplace equations with critical exponential growth in smooth bounded domain in ${\mathbb{R}^N}$ . Precisely we prove multibubble phenomena and obtain an energy inequality for those concentrating solutions. In fact we partly extend the corresponding two-dimensional results of Adimurthi and Struwe (J Funct Anal 175:125?C167, 2000) and Druet (Duke Math J 132:217?C269, 2006) to high dimensional case.     

Isometric Immersions into {\mathbb{S}^m \times \mathbb{R}} and {\mathbb{H}^m \times \mathbb{R}} with High Codimensions

In this paper, we obtain sufficient and necessary conditions for a simply connected Riemannian manifold (M ⁿ , g) to be isometrically immersed into ${\mathbb{S}^m \times \mathbb{R}}$ and ${\mathbb{H}^m \times \mathbb{R}}$ .     

On Quadratic Differentials and Twisted Normal Maps of Surfaces in {\mathbb{S}^{2} \times\mathbb{R}} and {\mathbb{H}^{2} \times\mathbb{R}}

Given a Lie group G with a bi-invariant metric and a compact Lie subgroup K, Bittencourt and Ripoll used the homogeneous structure of quotient spaces to define a Gauss map ${\mathcal{N}:M^{n}\rightarrow{\mathbb{S}}}$ on any hypersupersurface ${M^{n}\looparrowright G/K}$ , where ${{\mathbb{S}}}$ is the unit sphere of the Lie algebra of G. It is proved in Bittencourt and Ripoll (Pacific J Math 224:45–64, 2006) that M ⁿ having constant mean curvature (CMC) is equivalent to ${\mathcal{N}}$ being harmonic, a generalization of a Ruh–Vilms theorem for submanifolds in the Euclidean space. In particular, when n = 2, the induced quadratic differential ${\mathcal{Q}_{\mathcal{N}}:=(\mathcal{N}^{\ast}g)^{2,0}}$ is holomorphic on CMC surfaces of G/K. In this paper, we take ${G/K={\mathbb{S}}^{2}\times{\mathbb{R}}}$ and compare ${\mathcal{Q}_{\mathcal{N}}}$ with the Abresch–Rosenberg differential ${\mathcal{Q}}$ , also holomorphic for CMC surfaces. It is proved that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ , after showing that ${\mathcal{N}}$ is the twisted normal given by (1.5) herein. Then we define the twisted normal for surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ and prove that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ as well. Within the unified model for the two product spaces, we compute the tension field of ${\mathcal{N}}$ and extend to surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ the equivalence between the CMC property and the harmonicity of ${\mathcal{N}.}$      

Classification of rotational special Weingarten surfaces of minimal type in {\mathbb{S}^2 \times \mathbb{R}} and {\mathbb{H}^2 \times \mathbb{R}}

In this paper we classify the complete rotational special Weingarten surfaces in ${\mathbb{S}^2 \times \mathbb{R}}$ and ${\mathbb{H}^2 \times \mathbb{R}}$ ; i.e. rotational surfaces in ${\mathbb{S}^2 \times \mathbb{R}}$ and ${\mathbb{H}^2 \times \mathbb{R}}$ whose mean curvature H and extrinsic curvature K _e satisfy H = f(H ² ? K _e ), for some function ${f \in \mathcal{C}^1([0,+\infty))}$ such that f(0) = 0 and 4x(f′(x))² < 1 for any x ≥ 0. Furthermore we show the existence of non-complete examples of such surfaces.     

Existence of solutions for a perturbation sublinear elliptic equation in {\mathbb {R}^N}

In this paper we consider the following problem $\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.$ ${f \in L^2(\mathbb{R}^N)\cap L^\frac{2(1-\theta)}{1-2\theta}(\mathbb{R}^N),\, N\geq 3,\, f\geq 0,\, f \neq 0}In this paper we consider the following problem

{l -Du=u-|u|-2qu+f u ? H1(\mathbbRN)?L2(1-q)(\mathbbRN)\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.      8. Complexity of asymptotic behavior of the porous medium equation in {\mathbb{R}^N}      Jingxue Yin  Liangwei Wang  Rui Huang 《Journal of Evolution Equations》2011,11(2):429-455       9. Global attractor for a semilinear strongly degenerate parabolic equation on {\mathbb{R}^N}      Cung The Anh 《NoDEA : Nonlinear Differential Equations and Applications》2014,21(5):663-678 The aim of this paper is to prove the existence of the global attractor for a semilinear strongly degenerate parabolic equation on \({\mathbb{R}^N}\) with the locally Lipschitz nonlinearity satisfying a subcritical growth condition.      10. Three radial positive solutions for semilinear elliptic problems in $\mathbb{R}^N}          下载免费PDF全文 Suyun Wang  Yanhong Zhang  Ruyun Ma 《Journal of Applied Analysis & Computation》2020,10(2):760-770 This paper is concerned with the semilinear elliptic problem $$ \left\{ \begin{aligned} &-\Delta u=\lambda h(|x|)f(u) \ \ \ \ \ \ \ \ \ \ \text{in}\ \mathbb{R}^N, \\~ & u(x)>0\hskip 3cm \ \text{in}\ \mathbb{R}^N, \\~ &u\to 0 \hskip 3cm \ \ \ \ \text{as}\ |x|\to \infty, \end{aligned} \right. $$ where $\lambda$ is a real parameter and $h$ is a weight function which is positive. We show the existence of three radial positive solutions under suitable conditions on the nonlinearity. Proofs are mainly based on the bifurcation technique.      11. Asymmetric positive solutions for a symmetric nonlinear problem in {mathbb R}^N      Silvia Cingolani  José Luis Gámez 《Calculus of Variations and Partial Differential Equations》2000,11(1):97-117 We study a symmetric semilinear elliptic problem in all and we prove existence of an asymmetric positive solution by using variational arguments. The corresponding problem in dimension N=2, which provides the motivation of this work, arises in Nonlinear Optics from the study of the behaviour of optical cylindrical waveguides. Received September 28, 1999/ Accepted January 14, 2000 / Published online June 28, 2000      12. Existence and multiplicity of solutions of semi-linear elliptic equations in {mathbb R}^N      Boyan Sirakov 《Calculus of Variations and Partial Differential Equations》2000,11(2):119-142       13. Quasi-linear equations in $ \mathbb{R}^N $: perturbation results      Benedetta PELLACCI 《NoDEA : Nonlinear Differential Equations and Applications》2003,10(1):95-117       14. Hausdorff measures of subgroups of \mathbb{R }/\mathbb{Z } and \mathbb{R }      Hans Weber  Enrico Zoli 《Ricerche di matematica》2013,62(2):209-228 For a sequence $\underline{u}=(u_n)_{n\in \mathbb{N }}$ of integers, let $t_{\underline{u}}(\mathbb{T })$ be the group of all topologically $\underline{u}$ -torsion elements of the circle group $\mathbb{T }:=\mathbb{R }/\mathbb{Z }$ . We show that for any $s\in ]0,1[$ and $m\in \{0,+\infty \}$ there exists $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has Hausdorff dimension $s$ and $s$ -dimensional Hausdorff measure equal to $m$ (no other values for $m$ are possible). More generally, for dimension functions $f,g$ with $f(t)\prec g(t), f(t)\prec \!\!\!\prec t$ and $g(t)\prec \!\!\!\prec t$ we find $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has at the same time infinite $f$ -measure and null $g$ -measure.      15. A Construction of Partial Difference Sets in {\mathbb{Z}}_{p^{2 \times } } {\mathbb{Z}}_{p^{2 \times \cdot \cdot \cdot \times } } {\mathbb{Z}}_{p^2 }      Ka Hin Leung  Siu Lun Ma 《Designs, Codes and Cryptography》1996,8(1-2):167-172 In this paper, we give a construction of partial difference sets in p 2 x p 2 x ... x p 2using some finite local rings.Dedicated to Hanfried Lenz on the occasion of his 80th birthdayThe work of this paper was done when the authors visited the University of Hong Kong.      16. Log pairs on hypersurfaces of degreeN in {N{\text{ in }}\mathbb{P}^N      I. A. Chel’tsov 《Mathematical Notes》2000,68(1):113-119 The objective of this paper is to study the birational structure of smooth hypersurfaces of degreeN in by examining properties of moving log pairs on them. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp 131–138, July, 2000.      17. Rational holomorphic maps from \mathbb{B}^2 into \mathbb{B}^N with degree 2      Zhihua Chen  Shanyu Ji  Dekang Xu 《中国科学 数学(英文版)》2006,49(11):1504-1522 Rational proper holomorphic maps from the unit ball in ?2 into the unit ball ? N with degree 2 are studied. Any such map must be equivalent to one of the four types of maps.      18. Maximum distance separable codes over {\mathbb{Z}_4} and {\mathbb{Z}_2 \times \mathbb{Z}_4}      M. Bilal  J. Borges  S. T. Dougherty  C. Fern��ndez-C��rdoba 《Designs, Codes and Cryptography》2011,61(1):31-40 Known upper bounds on the minimum distance of codes over rings are applied to the case of ${\mathbb Z_{2}\mathbb Z_{4}}$ -additive codes, that is subgroups of ${\mathbb Z_{2}^{\alpha}\mathbb Z_{4}^{\beta}}$ . Two kinds of maximum distance separable codes are studied. We determine all possible parameters of these codes and characterize the codes in certain cases. The main results are also valid when ?? = 0, namely for quaternary linear codes.      19. Quasilinear elliptic equations on $$\mathbb{R}^N $$with infinitely many solutions      Patrick?J.?RabierEmail author 《NoDEA : Nonlinear Differential Equations and Applications》2004,11(3):311-333 We give verifiable conditions ensuring that second order quasilinear elliptic equations on have infinitely many solutions in the Sobolev space for generic right-hand sides. This amounts to translating in concrete terms the more elusive hypotheses of an abstract theorem. Salient points include the proof that a key denseness property is equivalent to the existence of nontrivial solutions to an auxiliary problem, and an estimate of the size of the set of critical points of nonlinear Schrödinger operators. Conditions for the real-analyticity of Nemytskii operators are also discussed.      20. A smooth global branch of solutions for a semilinear elliptic equation on {\mathbb{R}^N}      François Genoud 《Calculus of Variations and Partial Differential Equations》2010,38(1-2):207-232 The existence of a global branch of positive spherically symmetric solutions ${\{(\lambda,u(\lambda)):\lambda\in(0,\infty)\}}$ of the semilinear elliptic equation $$\Delta u - \lambda u + V(x)|u|^{p-1}u = 0 \quad \text{in}\,\mathbb{R}^N\,\text{with}\,N\geq3$$ is proved for ${1 < p < 1+\frac{4-2b}{N-2}}$ , where ${b\in(0,2)}$ is such that the radial function V vanishes at infinity like |x|?b . V is allowed to be singular at the origin but not worse than |x|?b . The mapping ${\lambda\mapsto u(\lambda)}$ is of class ${C^r((0,\infty),H^1(\mathbb{R}^N))}$ if ${V\in C^r(\mathbb{R}^N\setminus\{0\},\mathbb{R})}$ , for r = 0, 1. Further properties of regularity and decay at infinity of solutions are also established. This work is a natural continuation of previous results by Stuart and the author, concerning the existence of a local branch of solutions of the same equation for values of the bifurcation parameter λ in a right neighbourhood of λ = 0. The variational structure of the equation is deeply exploited and the global continuation is obtained via an implicit function theorem.

Global attractor for a semilinear strongly degenerate parabolic equation on {\mathbb{R}^N}

The aim of this paper is to prove the existence of the global attractor for a semilinear strongly degenerate parabolic equation on \({\mathbb{R}^N}\) with the locally Lipschitz nonlinearity satisfying a subcritical growth condition.     

Three radial positive solutions for semilinear elliptic problems in $\mathbb{R}^N}

This paper is concerned with the semilinear elliptic problem $$ \left\{ \begin{aligned} &-\Delta u=\lambda h(|x|)f(u) \ \ \ \ \ \ \ \ \ \ \text{in}\ \mathbb{R}^N, \\~ & u(x)>0\hskip 3cm \ \text{in}\ \mathbb{R}^N, \\~ &u\to 0 \hskip 3cm \ \ \ \ \text{as}\ |x|\to \infty, \end{aligned} \right. $$ where $\lambda$ is a real parameter and $h$ is a weight function which is positive. We show the existence of three radial positive solutions under suitable conditions on the nonlinearity. Proofs are mainly based on the bifurcation technique.     

Asymmetric positive solutions for a symmetric nonlinear problem in {mathbb R}^N

We study a symmetric semilinear elliptic problem in all and we prove existence of an asymmetric positive solution by using variational arguments. The corresponding problem in dimension N=2, which provides the motivation of this work, arises in Nonlinear Optics from the study of the behaviour of optical cylindrical waveguides. Received September 28, 1999/ Accepted January 14, 2000 / Published online June 28, 2000     

Hausdorff measures of subgroups of \mathbb{R }/\mathbb{Z } and \mathbb{R }

For a sequence $\underline{u}=(u_n)_{n\in \mathbb{N }}$ of integers, let $t_{\underline{u}}(\mathbb{T })$ be the group of all topologically $\underline{u}$ -torsion elements of the circle group $\mathbb{T }:=\mathbb{R }/\mathbb{Z }$ . We show that for any $s\in ]0,1[$ and $m\in \{0,+\infty \}$ there exists $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has Hausdorff dimension $s$ and $s$ -dimensional Hausdorff measure equal to $m$ (no other values for $m$ are possible). More generally, for dimension functions $f,g$ with $f(t)\prec g(t), f(t)\prec \!\!\!\prec t$ and $g(t)\prec \!\!\!\prec t$ we find $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has at the same time infinite $f$ -measure and null $g$ -measure.     

A Construction of Partial Difference Sets in {\mathbb{Z}}_{p^{2 \times } } {\mathbb{Z}}_{p^{2 \times \cdot \cdot \cdot \times } } {\mathbb{Z}}_{p^2 }

In this paper, we give a construction of partial difference sets in _p ² x _p ² x ... x _p ²using some finite local rings.Dedicated to Hanfried Lenz on the occasion of his 80th birthdayThe work of this paper was done when the authors visited the University of Hong Kong.     

Log pairs on hypersurfaces of degreeN in {N{\text{ in }}\mathbb{P}^N

The objective of this paper is to study the birational structure of smooth hypersurfaces of degreeN in by examining properties of moving log pairs on them. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp 131–138, July, 2000.     

Rational holomorphic maps from \mathbb{B}^2 into \mathbb{B}^N with degree 2

Rational proper holomorphic maps from the unit ball in ?² into the unit ball ? ^N with degree 2 are studied. Any such map must be equivalent to one of the four types of maps.     

Maximum distance separable codes over {\mathbb{Z}_4} and {\mathbb{Z}_2 \times \mathbb{Z}_4}

Known upper bounds on the minimum distance of codes over rings are applied to the case of ${\mathbb Z_{2}\mathbb Z_{4}}$ -additive codes, that is subgroups of ${\mathbb Z_{2}^{\alpha}\mathbb Z_{4}^{\beta}}$ . Two kinds of maximum distance separable codes are studied. We determine all possible parameters of these codes and characterize the codes in certain cases. The main results are also valid when ?? = 0, namely for quaternary linear codes.     

Quasilinear elliptic equations on $$\mathbb{R}^N $$with infinitely many solutions

We give verifiable conditions ensuring that second order quasilinear elliptic equations on have infinitely many solutions in the Sobolev space for generic right-hand sides. This amounts to translating in concrete terms the more elusive hypotheses of an abstract theorem. Salient points include the proof that a key denseness property is equivalent to the existence of nontrivial solutions to an auxiliary problem, and an estimate of the size of the set of critical points of nonlinear Schrödinger operators. Conditions for the real-analyticity of Nemytskii operators are also discussed.     

A smooth global branch of solutions for a semilinear elliptic equation on {\mathbb{R}^N}

The existence of a global branch of positive spherically symmetric solutions ${\{(\lambda,u(\lambda)):\lambda\in(0,\infty)\}}$ of the semilinear elliptic equation $$\Delta u - \lambda u + V(x)|u|^{p-1}u = 0 \quad \text{in}\,\mathbb{R}^N\,\text{with}\,N\geq3$$ is proved for ${1 < p < 1+\frac{4-2b}{N-2}}$ , where ${b\in(0,2)}$ is such that the radial function V vanishes at infinity like |x|^?b . V is allowed to be singular at the origin but not worse than |x|^?b . The mapping ${\lambda\mapsto u(\lambda)}$ is of class ${C^r((0,\infty),H^1(\mathbb{R}^N))}$ if ${V\in C^r(\mathbb{R}^N\setminus\{0\},\mathbb{R})}$ , for r = 0, 1. Further properties of regularity and decay at infinity of solutions are also established. This work is a natural continuation of previous results by Stuart and the author, concerning the existence of a local branch of solutions of the same equation for values of the bifurcation parameter λ in a right neighbourhood of λ = 0. The variational structure of the equation is deeply exploited and the global continuation is obtained via an implicit function theorem.