relativistic chern simons higgs vortex equations an existence theorem is established for the solutions to the non abelian relativistic chern simons higgs vortex equations over a doubly periodic domain when the gauge group g assumes the most general and important prototype form g su n mathematics subject classification 2010 35c08 35j50 35q70 81e13

sharp estimates for solutions of multi bubbles in compact riemann surfaces in this paper we consider a sequence of multibubble solutions u k of the equation 0 1 0 u k k he uk f m he u k d 1 0 in m where h is a c 2 positive function in a compact riemann surface m and k is a constant satisfying lim k k 8m for some positive integer m 1 we prove among other things that k 8m 2 m m j 1h 1 p k j 0 log h p k j 8m 2k p k j k j e k j o e k j where p k j are centers of the bubbles of u k and k j are the local maxima of u k after adding a constant this yields a uniform bound of solutions as k converges to 8m from below provided that 0 log h p k j 8m 2k p k j 0 it generalizes a previous result due to ding jost li and wang 18 and nolasco and tarantello 31 which says that any sequence of minimizers u k is uniformly bounded if k 0 for any maximum point p of the sum of 2 log h and the regular part of the green function where k is the gaussian curvature of m the analytic work of this paper is the first step toward computing the topological degree of 0 1 which was initiated by li 24

solutions to the master equations governing fractional vortices abstract by means of variational methods in this paper we establish sharp existence results for solutions of the master equations governing fractional multiple vortices in the doubly periodic situation the conditions for existence are both necessary and sufficient and give the upper bounds for the vortex numbers in terms of the size of the periodic cell domain in the planar situation there is no restriction on the vortex numbers in both situations the solutions are uniquely determined by the prescribed locations and the local winding numbers of the vortices

maximal solution of the liouville equation in doubly connected domains abstract in this paper we consider the liouville equation u 2 e u 0 with dirichlet boundary conditions in a two dimensional doubly connected domain we show that there exists a simple closed curve such that for a sequence n 0 and a sequence of solutions u n it holds u n log 1 n h where h is a harmonic function in and n 2 log 1 n e u n d x 8 c where c is a constant depending on the conformal class of only

on the positivity of the effective action in a theory of random surfaces it is shown that the functional s eta frac 1 24 pi int left frac 1 2 left nabla eta right 2 2 eta right d mu 0 defined onc functions on the two dimensional sphere satisfies the inequalitys 0 if is subject to the constraint int e eta 1 d mu 0 0 the minimums 0 is attained at the solutions of the euler lagrange equations the proof is based on a sharper version of moser trudinger s inequality due to aubin which holds under the additional constraint int e eta xd mu 0 0 this condition can always be satisfied by exploiting the invariance ofs under the conformal transformations ofs2 the result is relevant for a recently proposed formulation of a theory of random surfaces

the sphere covering inequality and its applications in this paper we show that the total area of two distinct surfaces with gaussian curvature equal to 1 which are also conformal to the euclidean unit disk with the same conformal factor on the boundary must be at least 4 pi in other words the areas of these surfaces must cover the whole unit sphere after a proper rearrangement we refer to this lower bound of total area as the sphere covering inequality the inequality and its generalizations are applied to a number of open problems related to moser trudinger type inequalities mean field equations and onsager vortices etc and yield optimal results in particular we prove a conjecture proposed by chang and yang acta math 159 3 4 215 259 1987 in the study of nirenberg problem in conformal geometry

the webster scalar curvature flow on cr sphere part ii abstract this is the first of two papers in which we prove some properties of the webster scalar curvature flow more precisely we establish the long time existence l p convergence and the blow up analysis for the solution of the flow as a by product we prove the convergence of the cr yamabe flow on the cr sphere the results in this paper will be used to prove a result of prescribing webster scalar curvature on the cr sphere which is the main result of the second paper

a singular sphere covering inequality uniqueness and symmetry of solutions to singular liouville type equations we derive a singular version of the sphere covering inequality which was recently introduced in gui and moradifam invent math https doi org 10 1007 s00222 018 0820 2 2018 suitable for treating singular liouville type problems with superharmonic weights as an application we deduce new uniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains as well as new self contained proofs of previously known results such as the uniqueness of spherical convex polytopes first established in luo and tian proc am math soc 116 4 1119 1129 1992 furthermore we derive new symmetry results for the spherical onsager vortex equation

the moser trudinger inequality on kahler einstein manifolds we prove the conjecture of tian on the strong form of the moser trudinger inequality for k ahler einstein manifolds with positive first chern class when there are no holomorphic vector fields and more generally when the setting is invariant under a maximal compact subgroup of the automorphism group

on a system of nonlinear elliptic equations arising in theoretical physics in this paper we study a system of nonlinear elliptic equations known as the vortex equations in 2 dimensions arising from the field theoretical descriptions of several models in physics when the underlying space is a closed surface we prove the existence and uniqueness of a solution under a necessary and sufficient condition when the space is r2 we establish the existence uniqueness and sharp decay estimates for a solution

the nirenberg problem and its generalizations a unified approach making use of integral representations we develop a unified approach to establish blow up profiles compactness and existence of positive solutions of the conformally invariant equations p sigma v kv frac n 2 sigma n 2 sigma on the standard unit sphere mathbb s n for all sigma in 0 n 2 where p sigma is the intertwining operator of order 2 sigma finding positive solutions of these equations is equivalent to seeking metrics in the conformal class of the standard metric on spheres with prescribed certain curvatures when sigma 1 it is the prescribing scalar curvature problem or the nirenberg problem and when sigma 2 it is the prescribing q curvature problem

singular mean field equations on compact riemann surfaces for a general class of elliptic pde s in mean field form on compact riemann surfaces with exponential nonlinearity we address the question of the existence of solutions with concentrated nonlinear term which in view of the applications are physically of definite interest in the model we also include the possible presence of singular sources in the form of dirac masses which makes the problem more difficult to attack

blow up analysis of a finsler liouville equation in two dimensions abstract in this paper we establish a blow up analysis for the finsler liouville equation q u v x e u in two dimensions here the operator q is the so called finsler laplacian which is a generalized laplacian as a byproduct we obtain an existence result

on conformal deformations of metrics onsn abstract on s n there is a naturally metric defined n th order conformal invariant operator p n associated with this operator is a so called q curvature quantity when two metrics are pointwise conformally related their associated operators together with their q curvatures satisfy the natural differential equations this paper is devoted to the question of which function can be a q curvature candidate this is the so called prescribing q curvature problem our main result is that if q is positive nondegenerate and the naturally defined mapping associated with q has nonzero degree then our problem has a solution this is the natural generalization of prescribing gaussian curvature on s 2 into s n

invariant conformal metrics on ⁿ in this paper we use the relationship between conformal metrics on the sphere and horospherically convex hypersurfaces in the hyperbolic space for giving sufficient conditions on a conformal metric to be radial under some constrain on the eigenvalues of its schouten tensor also we study conformal metrics on the sphere which are invariant by a k parameter subgroup of conformal diffeomorphisms of the sphere giving a bound on its maximum dimension moreover we classify conformal metrics on the sphere whose eigenvalues of the shouten tensor are all constant we call them isoparametric conformal metrics and we use a clas

moser trudinger inequalities for singular liouville systems in this paper we prove a moser trudinger inequality for the euler lagrange functional of general singular liouville systems on a compact surface we characterize the values of the parameters which yield coercivity for the functional hence the existence of energy minimizing solutions for the system and we give necessary conditions for boundedness from below we also provide a sharp inequality under assuming the coefficients of the system to be non positive outside the diagonal the proofs use a concentration compactness alternative poho aev type identities and blow up analysis

remarks on the existence problem of positive kahler einstein metrics on introduit un invariant m qui donne un critere pour l existence de metriques de kahler einstein

an abstract existence theorem at resonance and its applications abstract by using brouwer degree theory and a continuation theorem based on mawhin s coincidence degree an abstract existence theorem at resonance for operator equations is developed as applications of this result some sufficient conditions are given for the existence of periodic solutions of semilinear equations at resonance where the kernel of the linear part has dimension n n 2 the results due to nagle and sinkala are generalized improved and unified

the scalar curvature flow on s n perturbation theorem revisited for the problem of finding a geometry on s n for n 3 with a prescribed scalar curvature there is a well known result which is called the perturbation theorem it is due to chang and yang duke math j 64 27 69 1991 their key assumption is that the candidate f for the prescribed scalar curvature is sufficiently near the scalar curvature of the standard metric in the sup norm it is important to know how large that difference in sup norm can possibly be here we consider prescribing scalar curvature problem using the scalar curvature flow

a note on the problem of prescribing gaussian curvature on surfaces the problem of existence of conformal metrics with gaussian curvature equal to a given function k on a compact riemannian 2 manifold m of negative euler characteristic is studied let ko be any nonconstant function on m with max ko 0 and let ka ko a it is proved that there exists a a 0 such that the problem has a solution for k ka iff a e oo a moreover if a e 0 a then the problem has at least 2 solutions let m be a closed 2 dimensional smooth manifold and g be a riemannian metric on m let k denote the gaussian curvature of g if g e2ug is another riemannian metric conformal to g and has gaussian curvature k then it is well known that k e 2u k au where a is the laplacian of g given a function k e c m the problem of prescribing gaussian curvature asks whether one can find u e c m such that the metric g e2ug has the given k as its gaussian curvature obviously this is equivalent to the problem of solvability of the following elliptic equation 1 au k ke2u o onm if u is a solution of 1 then we have by integrating 1

concerning a type of sobolev inequality on the sphere in chang and yang 1 5 a type of sobolev inequality was proved for a class of functions on s 2 it played an important role in the proof of the existence by the min max method of a pointwise conformal metric on s 2 which has some given function as its scalar curvature in this paper we prove an analogous inequality for functions on s n n 3 from now on denotes the integral average on s i e f s fdx vol s for functions on s and x j j 1 2 n 1 denote orthogonal coordinates in r 1

non degeneracy and uniqueness of solutions to singular mean field equations on bounded domains abstract the aim of this paper is to complete the program initiated in 51 23 and then carried out by several authors concerning non degeneracy and uniqueness of solutions to mean field equations in particular we consider mean field equations with general singular data on non smooth domains the argument is based on the alexandrov bol inequality and on the eigenvalues analysis of linearized singular liouville type problems

the liouville equation with singular data a concentration compactness principle via a local representation formula abstract for a bounded domain r 2 we establish a concentration compactness result for the following class of singular liouville equations u e u 4 j 1 m j p j in where pj j 0 and pj denotes the dirac measure with pole at point pj j 1 m our result extends brezis merle s theorem comm partial differential equations16 1991 1223 1253 concerning solution sequences for the regular liouville equation where the dirac measures are replaced by lp data p 1 in some particular case we also derive a mass quantization principle in the same spirit of li shafrir indiana univ math j 43 1994 1255 1270 our analysis was motivated by the study of the bogomol nyi equations arising in several self dual gauge field theories of interest in theoretical physics such as the chern simons theory self dual chern simons theories lecture notes in physics vol 36 springer verlag berlin 1995 and the electroweak theory selected papers on gauge theory of weak and electromagnetic interactions world scientific singapore

classification of ternary extremal self dual codes of length 28 all 28 dimensional unimodular lattices with minimum norm 3 are known using this classification we give a classification of ternary extremal self dual codes of length 28 up to equivalence there are 6 931 such codes

a compactness result for periodic multivortices in the electroweak theory we derive a priori uniform bounds for solutions of an elliptic system of liouville type equations first analyzed by j spruck and y yang comm math phys 144 1992 1 yielding periodic multivortices in the classical electroweak theory of glashow salam weinberg our proof is based on a concentration quantization result in the same spirit of brezis merle comm partial differential equations 16 8 9 1991 1223 and li shafrir indiana univ math j 43 4 1994 1255 for mean field equations on riemannian compact 2 manifolds

a special class of stationary flows for two dimensional euler equations a statistical mechanics description we consider the canonical gibbs measure associated to an vortex system in a bounded domain at inverse temperature widetilde beta and prove that in the limitn widetilde beta n n 1 where 8 here denotes the vorticity intensity of each vortex the one particle distribution function n nx x converges to a superposition of solutions of the following mean field equation left begin array 20 c varrho beta x frac e beta psi mathop smallint limits lambda e beta psi delta psi varrho beta in lambda psi partial lambda 0 end array right

prescribing webster scalar curvature on cr manifolds of negative conformal invariants abstract in this paper we are interested in solving the following partial differential equation u r u f u 1 2 n on a compact strictly pseudo convex cr manifold m of dimension 2 n 1 with n 1 this problem naturally arises when solving the prescribing webster scalar curvature problem on m with the prescribed function f using variational techniques we prove several non existence existence and multiplicity results when the function f is sign changing

conformal metrics on the unit ball with prescribed mean curvature this paper focuses on the study of the prescribed mean curvature problem on the unit ball if the difference between the mean curvature candidate f and mean curvature of the standard metric in the supremum norm is sufficiently small then the existence of positive solutions of conformal mean curvature equation has been known the purpose of the paper is to investigate quantitatively how large that difference can be by using a flow method

prescribing q curvature problem on sn let pn be the n th order paneitz operator on sn n 3 we consider the following prescribing q curvature problem on sn pnu n 1 q x enuon sn where q is a smooth positive function on sn satisfying the following non degeneracy condition q 2 q 2 0 let g sn rn 1 be defined by g x q x q x we show that if q 0 is non degenerate and deg g g sn 0 then the above equation has a solution when n is even this has been established in our earlier work j wei x xu on conformal deformation of metrics on sn j funct anal 157 1998 292 325 when n is odd pn becomes a pseudo differential operator here we develop a unified approach to treat both even and odd cases the key idea is to write it as an integral equation and use liapunov schmidt reduction method

conformal metrics with prescribed gaussian curvature on s2 we consider on s 2 the problem of which functions k can be the scalar curvature of a metric conformal to the standard metric on s 2 we assume that k is a function of one variable and we obtain a necessary and sufficient condition for the problem to be solvable we also obtain several new sufficient conditions on k which are easy to check in order that the problem be solvable

moser trudinger and beckner onofri s inequalities on the cr sphere we derive sharp moser trudinger inequalities on the cr sphere the rst type is in the adams form for powers of the sublaplacian and for general spectrally dened operators on the space of cr pluriharmonic functions we will then obtain the sharp beckner onofri inequality for crpluriharmonic functions on the sphere and as a consequence a sharp logarithmic hardy littlewood sobolev inequality in the form given by carlen and loss

on the perturbed q curvature problem on s4 abstract let g 0 denote the standard metric on s 4 and p g 0 g 0 2 2 g 0 denote the corresponding paneitz operator in this work we study the following fourth order elliptic problem with exponential nonlinearity p g 0 u 6 2 q x e 4 u on s 4 here q is a prescribed smooth function on s 4 which is assumed to be a perturbation of a constant we prove existence results to the above problem under assumptions only on the shape of q near its critical points these are more general than the non degeneracy conditions assumed so far we also show local uniqueness and exact multiplicity results for this problem the main tool used is the lyapunov schmidt reduction

generic multiplicity for a scalar field equation on compact surfaces we prove generic multiplicity of solutions for a scalar field equation on compact surfaces via morse inequalities in particular our result improves significantly the multiplicity estimate which can be deduced from the degree counting formula in chen and lin 2003 12 related results are derived for the prescribed q curvature equation

uniqueness of stable capillary hypersurfaces in a ball in this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical our result also provides a proof of a conjecture proposed by sternberg and zumbrun j reine angew math 503 63 85 1998 we also prove a heintze karcher ros type inequality for hypersurfaces with free boundary in a ball which together with the new minkowski formula yields a new proof of alexandrov s theorem for embedded cmc hypersurfaces in a ball with free boundary

extremal metrics of zeta function determinants on 4 manifolds in conformal geometry the sobolev inequality at a critical exponent has received much attention in particular the determination of the best constants has played a crucial role in the yamabe problem in dimension two the analogous problem deals with the moser trudinger inequality on a compact riemann surface m2 there exists a constant c c m so that f e47w2 c m if f ivw12 1 and f w 0 the connection of this inequality with geometry comes through the zeta functional determinant of the laplacian as defined by ray singer for a riemannian metric g let 0 a1 a2 be the spectrum

regular solutions to a supercritical elliptic problem in exterior domains we consider the supercritical elliptic problem u eu 0 in an exterior domain rn d under zero dirichlet condition where d is smooth and bounded in rn n 3 we prove that for small this problem admits infinitely many regular solutions

statistical mechanics approach to some problems in conformal geometry a weak law of large numbers is established for a sequence of systems of n classical point particles with logarithmic pair potential in rn or sn n n which are distributed according to the configurational microcanonical measure e h or rather some regularization thereof where h is the configurational hamiltonian and e the configurational energy when n with non extensive energy scaling e n2e the particle positions become i i d according to a self consistent boltzmann distribution respectively a superposition of such distributions the self consistency condition in n dimensions is some nonlinear elliptic pde of order n pseudo pde if n is odd with an exponential nonlinearity when n 2 this pde is known in statistical mechanics as poisson boltzmann equation with applications to point vortices 2d coulomb and magnetized plasmas and gravitational systems it is then also known in conformal differential geometry where it is the central equation in nirenberg s problem of prescribed gaussian curvature for constant gauss curvature it becomes liouville s equation which also appears in two dimensional so called quantum liouville gravity the pde for n 4 is paneitz equation and while it is not known in statistical mechanics it originated from a study of the conformal invariance of maxwell s electromagnetism and has made its appearance in some recent model of four dimensional quantum gravity in differential geometry the paneitz equation and its higher order n generalizations have applications in the conformal geometry of n manifolds but no physical applications yet for general n interestingly though all the paneitz equations have an interpretation in terms of statistical mechanics

on prescribing geodesic curvature on d2 abstract some new results are obtained for the problem of prescribing geodesic curvature k on d when k possesses some kinds of symmetries

on extremal self dual ternary codes of lengths 28 to 40 the extremal self dual ternary codes of lengths 28 32 and 36 with monomial automorphisms of prime order r or 5 and of length 40 with monomial automorphisms of prime order r 5 are enumerated for each length and prime considered all inequivalent extremal codes with an automorphism of that order are found

optimal control in prescribing webster scalar curvatures on 3 dimensional pseudo hermitian manifolds abstract in this work we give new existence and multiplicity results for the solutions of the prescription problem for the webster scalar curvature on a 3 dimensional pseudo hermitian manifold the critical points of prescribed functions verify mixed conditions we establish some morse inequalities at infinity and a poincare hopf type formula to give a lower bound on the number of solutions as well as an upper bound for the morse index of such solutions

on the best constant in the moser onofri aubin inequality let s2 be the 2 dimensional unit sphere and let j denote the nonlinear functional on the sobolev space h1 s2 defined by j alpha u frac alpha 16 pi int s 2 nabla u 2 d mu 0 frac 1 4 pi int s 2 u d mu 0 rm ln int s 2 e u frac d mu 0 4 pi where d mu 0 sin theta d theta wedge d phi onofri had established that j is non negative on h1 s2 provided 1 in this note we show that if j is restricted to those u in h 1 s 2 that satisfies the aubin condition int s 2 e u x j d mu 0 0 quad rm for all 1 leq j leq 3 then the same inequality continues to hold i e j u 0 whenever alpha geq frac 2 3 epsilon 0 for some epsilon 0 0 the question of chang yang on whether this remains true for all alpha geq frac 1 2 remains open

analytic aspects of the toda system i a moser trudinger inequality in this paper we analyze solutions of the open toda system and establish an optimal moser trudinger type inequality for this system let be a closed surface with area 1 and k aij n n the cartan matrix for su n 1 i e we show that has a lower bound in h1 n if and only if this inequality is optimal as a direct consequence if mj for 4 for j 1 2 n m has a minimizer u that satisfies 2001 john wiley sons inc

a quantization property for blow up solutions of singular liouville type equations abstract motivated by the study of self dual vortices in gauge field theory cf vortices and monopoles birkhauser boston 1980 solitons in field theory and nonlinear analysis monographs in mathematics springer new york 2001 we consider a concentrating solution sequence u k satisfying u k z 2 k v k z e u k in b 1 z 2 k v k z e u k z 0 weakly in the sense of measures in b 1 where i k r k ii v k c 0 1 b 1 0 a v k b and v k a in b 1 we prove that necessarily 8 n 8 1 8 z the result is sharp as shown by explicit examples and should be compared with that obtained by li shafrir ind univ math j 43 4 1994 1255 concerning the situation where k 0 k n

sharp hardy littlewood sobolev inequalities on quaternionic heisenberg groups this paper is the second one following christ et al nonlinear anal 130 361 395 2016 in a series considering sharp hardy littlewood sobolev inequalities on groups of heisenberg type the first important breakthrough was made in frank et al ann math 176 349 381 2012 in this paper analogous results are obtained for the octonionic heisenberg group

scalar curvature equation on sn part i topological conditions abstract this is the first part of a series devoting to the study of the prescribing scalar curvature problem on the standard sphere of any dimension in the first part we will adopt the degree theoretic approach to give a topological condition and some general explicit conditions on the scalar curvature functions to ensure the solvability of the problem our topological condition is imposed on some of simple maps explicitly defined by the scalar curvature function which is derived from the asymptotic expansion of the boundary map introduced in a chang p yang a perturbation result in prescribing scalar curvature on s n duke math j 64 1991 27 69 our conditions particularly allowing non isolation and non degeneracy of the critical points of the scalar curvature functions can be easily verified in many situations in the second part of series we will make a detailed study on the verification of the topological condition our results will generalize almost all existing ones in the same direction and meanwhile provide a unified treatment for both symmetric and non symmetric cases of the scalar curvature functions

on ternary self dual codes of length 24 a partial classification is given of the self dual codes of length 24 over gf 3 the main results are as follows there are exactly two codes with minimum hamming distance d 9 most of the codes have d 6 and are indecomposable one code with d 6 has a trivial automorphism group this is the first such self dual code that has been found the codes generated by the 59 inequivalent 24 times 24 hadamard matrices have been investigated and there appear to be only nine inequivalent codes two with d 9 and seven with d 6 and in all there are 27 decomposable codes at least 96 indecomposable codes with d 6 and the total number of inequivalent codes is at least 140

crc handbook of combinatorial designs balanced incomplete block designs and t designs2 v k l designs of small orderbibds with small block sizet designs t 3steiner systemssymmetric designsresolvable and near resolvable designslatin squares mols and orthogonal arrayslatin squaresmutually orthogonal latin squares mols incomplete molsorthogonal arrays of index more than oneorthogonal arrays of strength more than twopairwise balanced designspbds and gdds the basicspbds recursive constructionspbd closurepairwise balanced designs as linear spacespbds and gdds of higher indexpbds frames and resolvabilityother combinatorial designsassociation schemesbalanced part ternary designsbalanced tournament designsbhaskar rao designscomplete mappings and sequencings of finite groupsconfigurationscostas arrayscoveringscycle systemsdifference familiesdifference matricesdifference sets abeliandifference sets nonabeliandifference triangle setsdirected designsd optimal matricesembedding partial quasigroupsequidistant permutation arraysfactorial designsfrequency squaresgeneralized quadranglesgraph decompositions and designsgraphical designshadamard matrices and designshall triple systemshowell designsmaximal sets of molsmendelsohn designsthe oberwolfach problemordered designs and perpendicular arraysorthogonal designsorthogonal main effect planspackingspartial geometriespartially balanced incomplete block designsquasigroupsquasi symmetric designs r l designsroom squaresself orthogonal latin squares sols sols with a symmetric orthogonal mate solssom sequences with zero autocorrelationskolem sequencesspherical t designsstarterstrades and defining sets t m s netstuscan squarest wise balanced designsuniformly resolvable designsvector space designsweighing matrices and conference matriceswhist tournamentsyouden designs generalizedyouden squaresapplicationscodescomputer science selected applicationsapplications of designs to cryptographyderandomizationoptimality and efficiency comparing block designsgroup testingscheduling a tournamentwinning the lotteryrelated mathematics and computational methodsfinite groups and designsnumber theory and finite fieldsgraphs and multigraphsfactorizations of graphsstrongly regular graphstwo graphsclassical geometriesprojective planes nondesarguesiancomputational methods in design theoryindex

kahler einstein metrics with positive scalar curvature in this paper we prove that the existence of kahler einstein metrics implies the stability of the underlying kahler manifold in a suitable sense in particular this disproves a long standing conjecture that a compact kahler manifold admits kahler einstein metrics if it has positive first chern class and no nontrivial holomorphic vector fields we will also establish an analytic criterion for the existence of kahler einstein metrics our arguments also yield that the analytic criterion is satisfied on stable kahler manifolds provided that the partial c 0 estimate posed in t6 is true

on the existence of positive periodic solutions for neutral functional differential equation with multiple deviating arguments by means of the abstract continuation theory for k contractions some criteria are established for the existence and nonexistence of positive periodic solutions of the following neutral functional differential equation dndt n t a t t n t j 1nbj t nt j t i 1mci t n t i t

a small note on symmetric geodesic curvature on d2 some new results are obtained for the problem of prescribing geodesic curvature k on d when k possesses some kinds of symmetries

topological degree for a mean field equation on riemann surfaces we consider the following mean field equations 0 1 where m is a compact riemann surface with volume 1 h is a positive continuous function on m is a real number 0 2 and where is a bounded smooth domain h is a c1 positive function on and based on our previous analytic work 14 we prove among other things that the degree counting formula for 0 1 is given by for 8m 8 m 1 2003 wiley periodicals inc

remarks on prescribing gauss curvature we study the nonlinear partial differential equation for the problem of prescribing gauss curvature k on s 2 we give an example of a rotationally symmetric k for which the kazdan warner obstruction is satisfied but the equation has no rotationally symmetric solution on the other hand we give a simple sufficient condition for solvability of the equation when k is rotationally symmetric finally we give a sufficient condition for solvability when k is not necessarily rotationally symmetric

blow up analysis and existence results in the supercritical case for an asymmetric mean field equation with variable intensities abstract a class of equations with exponential nonlinearities on a compact riemannian surface is considered more precisely we study an asymmetric sinh gordon problem arising as a mean field equation of the equilibrium turbulence of vortices with variable intensities we start by performing a blow up analysis in order to derive some information on the local blow up masses as a consequence we get a compactness property in a supercritical range we next introduce a variational argument based on improved moser trudinger inequalities which yields existence of solutions for any choice of the underlying surface

existence of periodic solutions for a kind of second order neutral functional differential equation by means of the continuation theorem of coincidence degree theory we study a kind of second order neutral functional differential equation as followsd 2dt 2u t j 1 nc ju t r j f u t u t a t g u t j 1 n b j t g u t c j t p t some new results on the existence of periodic solutions are obtained