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

complete classification of shape functions of self similar solutions abstract in this paper we study some asymptotic profiles of shape functions of self similar solutions to the initial boundary value problem with neumann boundary condition for the generalized kpz equation u t u x x u x q where q is positive number the shapes of solutions of the corresponding nonlinear ordinary differential equation are of very different nature the properties depend on the critical value q 1 3 2 2 and initial data as usual

a porous media equation with absorption ii uniqueness of the very singular solution abstract we prove the uniqueness of the very singular solution for an equation of the form formula where 1 2 n m p m 2 n the solution we find is of the form formula where f is the unique nontrivial solution of an ordinary differential equation formula with conditions f 0 on 0 f 0 0 and lim n 2 p m f 0

an improved geometric inequality via vanishing moments with applications to singular liouville equations we consider a class of singular liouville equations on compact surfaces motivated by the study of electroweak and self dual chern simons theories the gaussian curvature prescription with conical singularities and onsager s description of turbulence we analyse the problem of existence variationally and show how the angular distribution of the conformal volume near the singularities may lead to improvements in the moser trudinger inequality and in turn to lower bounds on the euler lagrange functional we then discuss existence and non existence results

asymptotic behaviors of governing equation of gauged sigma model for heisenberg ferromagnet abstract in this note we study weak solutions of equation 0 1 u 4 e u 1 e u 4 i 1 n p i 4 j 1 m q j in r 2 where p i i 1 n resp q j j 1 m are dirac masses concentrated at the points p i i 1 n resp q j j 1 m and n m 1 eq 0 1 represents a governing equation of gauged sigma model for heisenberg ferromagnet we show that it has a sequence of solutions u having behaviors as ln x o 1 at infinity with a free parameter 2 2 n m and our concern in this paper is to study the asymptotic behavior of b as approaching the extremal values 2 and 2 n m

computational comparison on the partitioning strategies in multiple choice integer programming abstract in this paper we review the theoretical background of two partitioning strategies the weighted mean method wmm and the reformulation and transformation technique rtt incorporated in the special ordered set branch and bound procedures leading to the global optimum in multiple choice integer programming mcip procedure flow is developed with a numerical example presented to illustrate the effect of these two partitioning strategies in the algorithm this procedure flow is then coded using ibm apl2 a total of 24 test problems are solved by this apl2 code for mcip to analyze the performance of wmm and rtt in mcip the preliminary computational results indicate that on the average rtt produces smaller branching trees than does wmm its performance tends to be better for those mcip problems having more multiple choice sets and a fewer average number of variables in each set however the average cpu time consumed by using rtt does not differ much from that consumed by using wmm

blow up analysis for boltzmann poisson equation in onsager s theory for point vortices with multi intensities abstract in this paper we consider the minimizing sequence for some energy functional of an elliptic equation associated with the mean field limit of the point vortex distribution one sided borel probability measure if such a sequence blows up we derive some estimate which is related to the behavior of solution near the blow up point moreover we study the two intensities case to consider the sufficient condition for this estimate our main results are new for the standard mean field equation as well

two dimensional solutions of a mean field equation on flat tori abstract we study the mean field equation on the flat torus t c z z u e u t e u 1 t 0 where is a real parameter for a general flat torus we obtain the existence of two dimensional solutions bifurcating from the trivial solution at each eigenvalue up to a multiplicative constant t of laplace operator on the torus in the space of even symmetric functions we further characterize the subset of all eigenvalues through which only one bifurcating curve passes finally local convexity near bifurcating points of the solution curves are obtained

an ideal column algorithm for integer programs with special ordered sets of variables one of the most frequently occurring integer programming structures is the one which has special ordered sets of variables included in multiple choice constraints for problems with this structure a set of ideal columns are defined from the linear programming relaxation of the integer program and a reduced integer program is formed by keeping only those columns within a specified distance from the ideal column conditions are established which guarantee when the optimal solution to the reduced problem is als optimal for the original problem when these conditions are not satisfied bounds on the optimal solution value are provided ideal columns are also used to establish weights for the special ordered set variables this procedure has been implemented through a control program written by the authors for mpsx 370 mip 370 computational results are given

on the solutions of liouville systems for some prescribed m 0 together with the dirichlet boundary condition u 0 on 0 the liouville equation finds applications in various fields of physics and mathematics to name a few it represents the newtonian potential of a cluster of self gravitating mass distribution where a 0 1 6 36 37 the electric potential induced by charge carriers in the theory of electrolytes where a 0 31 and maximal entropy solutions of the incompressible euler equation 23 9 in the field of differential geometry it stands for the problem of finding a metric corresponding to a prescribed gaussian curvature 3 10 the system analog of l takes the form

on anisotropic singularities for semi linear elliptic equations in r2 abstract in this note we study weak solutions of 1 u g u 8 0 x 2 in r 2 g u x 0 as x where g s 2 e s 1 e s 1 is an odd increasing and bounded function and 0 x 2 is defined in the sense of distribution 0 x 2 0 x 2 for c c 1 r 2 we obtain that problem 1 admits a unique x 2 odd weak solution which has anisotropic singularity 4 x 2 x 2 near the origin

qualitative properties of solutions to some nonlinear elliptic equations in r 2 o introduction elliptic equations in this paper we investigate properties of the solutions to the au r x e u x x r 2 for functions r x which are positive near infinity equations of this kind arise from a variety of situations such as from prescribing gaussian curvature in geometry i 2 and from combustion theory in physics 3 recently a series ofworks have been sought to understand the existence and the qualitative properties of the solutions of ni i 5 and ni cheng 4 1 considered the case where r x is nonpositive mcowen 6 1 and aviles 7 investigated the situation where r x 0 in some order as ixl in our previous paper 1 we consider a special case where r is a constant we proved that the solutions are radially symmetric and hence classified all the solutions in this paper we consider more general functions r x first we obtain the asymptotic behavior of the solution near infinity consequently we prove that all the solutions satisfy an identity which is somewhat of a generalization of the well known kazdan warner condition finally using the asymptotic behavior together with the further development of the method employed in our previous paper i 1 we show that all the solutions are radially symmetric provided r is radially symmetric and nonincreasing this part can be viewed as the completion of 1 throughout this paper we assume that the function r x is positive near infinity in 1 we study the asymptotic behavior of the solution u x of let fl 1 2n r2 r x e dx under some appropriate conditions we show that the solutions approach

a very singular solution of the porous media equation with absorption turns out to play an important role in the description of both the short and the long time behaviour of positive solutions of eq 1 1 for the short time behaviour it provides together with the one parameter family of singular solutions i of problem i a complete description of the possible types of singular behaviour at the point 0 o 121 for the long time behaviour of solutions u x t of the cauchy problem for 1 1 1 2 it provides the asymptotic form when the initial function decays sufficiently fast as 1x1 co t p u x t w x t 0 ast co uniformly in r 7 81 in this paper we shall show that if

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

stabilization of solutions of a degenerate nonlinear diffusion problem i 4 u f u in l l x r u tl t o in r u x 0 uo x in l l where m 1 is a parameter f is locally lipschitz continuous f 0 0 and u is bounded problems of this form arise in a number of areas of science for instance in models for gas or fluid flow in porous media 2 and for the spread of certain biological populations 13 161 this paper is divided into two parts in part i we consider what may be called the motivating example problem i which consists of problem i with the special choice f u u 1 u u a 1 for suitably restricted parameters a we begin by describing in detail the set l of nonnegative equilibrium solutions of problem i clearly 8 l contains the trivial solution u 0 for all l 0 write 8 l e l o in the description of z l there are two critical parameter values lo and l1 with 0 lo l1 j we show that i g l 4 for 0 l lo

conformal deformation of metrics on s 2 soit d un domaine lisse de s 2 et k une fonction lisse definie sur d alors pour que le probleme aux limites de neumann u k exp 2u 1 sur d u n 0 sur d ait une solution u on a a quand aire d max x d k x 0

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

asymptotic blow up analysis for singular liouville type equations with applications abstract we generalize the pointwise estimates obtained in 2 19 and 34 concerning blow up solutions of the liouville type equation u x 2 w x e u in with r 2 open and bounded 1 and w any lipschitz continuous function which satisfies 0 a w b we focus to the case left open in 2 and 34 where the parameter n whose analysis is much more involved as we need to resolve the difficulty of a genuinely non radial behaviour of blow up solutions in the worst situation there is no chance in general to resolve the profile in the form of a solution of a liouville equation in r 2 instead we need to adopt iterated blow up arguments next we refine our blow up analysis to cover a class of planar liouville type problems see 1 27 1 28 below arising from the study of cosmic strings cfr 28 35 in this context we are able to distinguish between a single blow up radial profile and the case of multiple blow up profiles typical of non radial solutions as a consequence we obtain a radial symmetry result which is interesting in itself but also contributes towards the sharp solvability issue for the planar problem 1 27 1 28

classification of solutions of elliptic equations arising from a gravitational o 3 gauge field model abstract in this paper we study an elliptic equation arising from the self dual maxwell gauged o 3 sigma model coupled with gravity when the parameter equals 1 and there is only one singular source we consider radially symmetric solutions there appear three important constants a positive parameter a representing a scaled gravitational constant a nonnegative integer n 1 representing the total string number and a nonnegative integer n 2 representing the total anti string number the values of the products a n 1 a n 2 0 play a crucial role in classifying radial solutions by using the decay rates of solutions at infinity we provide a complete classification of solutions for all possible values of a n 1 and a n 2 this improves previously known results

existence and uniqueness of topological multivortex solutions of the self dual chern simons cp 1 model abstract in this paper we study the existence and uniqueness of the topological chern simons vortices in the cp 1 model in r 2 after reducing the self dual equations to semilinear elliptic partial differential equations we show that a topological solution exists and it is unique up to a real smooth function if the chern simons coupling constant is sufficiently small

the asymptotic self similar behavior for the quasilinear heat equation with nonlinear boundary condition in this paper the quasilinear heat equation with the nonlinear boundary condition is studied the blow up rate and existence of a self similar solution are obtained it is proved that the rescaled function v y t t t 1 2 p a 2 u t t p 1 2 p a 2 y t behaves as t t like a nontrivial self similar profile

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

the solution structure of the o 3 sigma model in a maxwell chern simons theory in this paper a system of semilinear elliptic equations arising from a relativistic self dual maxwell chern simons o 3 sigma model is considered we reveal the uniqueness aspect of the topological solutions for the model the uniqueness result is associated with a clear solution structure of the equations of the radially symmetric case we locate each solution set denoted by a planar diagram

new blow up phenomena for su n 1 toda system abstract we consider the su n 1 toda system s u 1 2 e u 1 e u 2 e u k 0 in u 2 e u 1 2 e u 2 e u k 0 in u k e u 1 e u 2 2 e u k 0 in u 1 u 2 u k 0 on if 0 and is symmetric with respect to the origin we construct a family of solutions u 1 u k to s such that the i th component u i blows up at the origin with a mass 2 i 1 as goes to zero

evaluating solutions on an elliptic problem in a gravitational gauge field theory abstract an elliptic equation arising from the study of static solutions with prescribed zeros and poles of the einstein equations coupled with the classical sigma model and an abelian gauge field is considered we classify the solutions and establish the uniqueness of radially symmetric solutions we also complete a classification of symmetric solutions of an elliptic equation on the sphere

uniqueness and symmetry results for solutions of a mean field equation on 2 via a new bubbling phenomenon motivated by the study of gauge field vortices we consider a mean field equation on the standard two sphere involving a dirac distribution supported at a point p of s 2 as needed for the applications we show that solutions concentrate exactly at p for some limiting value of a given parameter we use this fact to obtain symmetry about the axis op and uniqueness property for the solution the presence of the dirac measure makes such a task particularly delicate indeed we need to rule out the possibility that after blow up in a suitable scale the solution sequence may admit a double peak profile described in terms of appropriate limiting problems for instance we need to account for the presence of non axially symmetric solutions in such limiting problems in this process we establish a symmetry result about a maximal circle through p and its antipodal point p that applies to more general situations where the full axial symmetry cannot be expected

moving planes moving spheres and a priori estimates abstract in this paper we use the method of moving planes and the method of moving spheres to obtain a priori estimates for the solutions of semi linear elliptic equations using the method of moving planes we establish a sharper estimate on the solutions for prescribing scalar curvature equations with indefinite curvature functions and thus generalized a resent result of lin applying the method of moving spheres we give a different proof for a well known sup inf inequality established by brezis li and shafrir

bubbling nodal solutions for a large perturbation of the moser trudinger equation on planar domains in this work we study the existence of nodal solutions for the problem begin aligned delta u lambda u e u 2 u p quad text in quad omega quad u 0 quad text on partial omega end aligned where omega subseteq mathbb r 2 is a bounded smooth domain and p rightarrow 1 if omega is a ball it is known that the case p 1 defines a critical threshold between the existence and the non existence of radially symmetric sign changing solutions in this work we construct a blowing up family of nodal solutions to such problem as p rightarrow 1 when omega is an arbitrary domain and lambda is small enough as far as we know this is the first construction of sign changing solutions for a moser trudinger critical equation on a non symmetric domain

large time behaviour of the solutions of a semilinear parabolic equation in rn abstract the asymptotic behaviour as t tends to of the solution of u t u u u p 1 0 in r n r p 1 was studied it was proved that the behaviour depends strongly on the sign of n 2 n p and also on the rate of decay of the admissible initial data u 0 x as x tends to

blow up analysis for a cosmic strings equation abstract in this paper we develop a blow up analysis for solutions of an elliptic pde of liouville type over the plane such solutions describe the behavior of cosmic strings parallel in a given direction for a w boson model coupled with einstein s equation we show how the blow up behavior of the solutions is characterized according to the physical parameters involved by new and surprising phenomena for example in some cases after a suitable scaling the blow up profile of the solution is described in terms of an equations that bares a geometrical meaning in the context of the uniformization of the riemann sphere with conical singularities

periodic maxwell chern simons vortices with concentrating property in order to study electrically and magnetically charged vortices in fractional quantum hall effect and anyonic superconductivity the maxwell chern simons mcs model was introduced by lee et al phys lett b 252 79 83 1990 as a unified system of the classical abelian higgs model ah and the chern simons cs model in this article the first goal is to obtain the uniform cs limit result of mcs model with respect to the chern simons parameter without any restriction on either a particular class of solutions or the number of vortex points as the chern simons mass scale tends to infinity the most important step for this purpose is to derive the relation between the higgs field and the neutral scalar field our cs limit result also provides the critical clue to answer the open problems raised by ricciardi and tarantello comm pure appl math 53 811 851 2000 and tarantello milan j math 72 29 80 2004 and we succeed to establish the existence of periodic maxwell chern simons vortices satisfying the concentrating property of the density of superconductive electron pairs furthermore we expect that the cs limit analysis in this paper would help to study the stability multiplicity and bubbling phenomena for solutions of the mcs model

branch and bound methods for multi item scheduling this paper presents two solution approaches for the multi item activity scheduling problem a problem of practical importance because it models many resource allocation and combinatorial problems both approaches are easily incorporated into commercially available linear programming lp based branch and bound codes such as ibm s mpsx370 mip many practitioners rely on such codes and will find these approaches useful the first approach is a specialization of a method for decomposing integer programs with block angular structure in the second approach the reduced costs in the solution to the usual lp relaxation are used to develop improved priorities for branching computational results on real applications are reported using mpsx 370 mip

a very singular solution for the slow diffusion equation with nonlinear convection abstract we here investigate an existence and uniqueness of the nontrivial nonnegative solution of a nonlinear ordinary differential equation f m r f f f q 0 satisfying a specific decay rate lim r r f r 0 with 1 2 q m 1 and q m 2 q m 1 here m 1 m q m 1 and 1 or 1 such a solution arises naturally when we study a very singular solution for a slow diffusion equation with nonlinear convection u t u m x x u q x defined either on the whole real line or on the half line 0

multiplicity results for the assigned gauss curvature problem in r2 to study the problem of the assigned gauss curvature with conical singularities on riemanian manifolds we consider the liouville equation with a single dirac measure on the two dimensional sphere by a stereographic projection we reduce the problem to a liouville equation on the euclidean plane we prove new multiplicity results for bounded radial solutions which improve on earlier results of c s lin and his collaborators based on numerical computations we also present various conjectures on the number of unbounded solutions using symmetries some multiplicity results for non radial solutions are also stated

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

existence of topological multi string solutions in abelian gauge field theories in this paper we consider a general form of self dual equations arising from abelian gauge field theories coupled with the einstein equations by applying the super subsolution method we prove that topological multi string solutions exist for any coupling constant which improves previously known results we provide two examples for application the self dual einstein maxwell higgs model and the gravitational maxwell gauged o 3 sigma model

uniqueness of the topological multivortex solution in the self dual chern simons theory we establish a uniqueness result for the topological multivortex solution to the self dual equations of the abelian relativistic self dual chern simons higgs model we prove that the topological multivortex solution is unique if the chern simons coupling parameter 0 is sufficiently small we also establish a uniqueness result for 0 sufficiently large

on the structure of the conformal gaussian curvature equation on r2 ii in this paper we continue our investigation initiated in ecn on the structure of the set of all possible solutions of the equation 1 1 au ke2 o in r 2 where a is the laplace operator and k is a given nonpositive locally hslder continuous function in r 2 equation 1 1 arises in the problem of finding a riemannian metric which realizes the given function k as its gaussian curvature and is conformal to the standard euclidean metric in r 2 we refer the reader to cn for a brief description of the background and the history of this problem the study of the equation 1 1 dates back to at least 1930 s but it seems that a complete classification of all solutions of 1 1 was obtained only very recently in cn for the case k x ix t near oo where l 2 is a constant here we continue to use the notation g near to denote that there exist two positive constants cl c2 such that c l f g c2f near oo as we did in cn more precisely the following result was established in cn theorem suppose that k 2 then the following conclusions hold i for each 0 1 1 possesses a unique solution u such that

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

degree counting for toda system with simple singularity one point blow up abstract in this paper we study the degree counting formula of the rank two toda system with simple singular source when 1 0 4 4 8 and 2 4 n the key step is to derive the degree formula of the shadow system which arises from the bubbling solutions as 1 tends to 4 in order to compute the topological degree of the shadow system we need to find some suitable deformation during this deformation we shall deal with new difficulty arising from the phenomenon blow up does not necessarily imply concentration of mass this phenomenon occurs due to the collapsing of singularities this is a continuation of the previous work 25

a porous media equation with absorption i long time behaviour abstract we study the long time behaviour of nonnegative solutions of a porous media equation with absorption formula formula where m 1 2 n max 1 m p m 2 n and x is a given bounded nonnegative function satisfying formula where 2 p m it is in particular shown that the long time behaviour is determined by the unique nonnegative solution of 1 with an initial function formula

a sup inf inequality near r 0 let be a domain in r2 not necessarily bounded consider the semi linear elliptic equation u r x eu x we prove that for any compact subset k of there is a constant c such that the inequality 1 supku inf u c holds for all solutions u this type of inequality was first established by brezis li and shafrir h brezis y y li i shafrir a sup inf inequality for some nonlinear elliptic equations involving exponential nonlinearities j funct anal 115 1993 344 358 under the assumption that r x is positive and bounded away from zero it has become a useful tool in estimating the solutions of semi linear elliptic equations either in euclidean spaces or on riemannian manifolds see h brezis y y li i shafrir a sup inf inequality for some nonlinear elliptic equations involving exponential nonlinearities j funct anal 115 1993 344 358 c c chen c s lin a sharp sup inf inequality for a nonlinear elliptic equation in r2 comm anal geom 6 1998 1 19 w chen c li gaussian curvature in the negative case proc amer math soc 131 2003 741 744 w chen c li indefinite elliptic problems with critical exponent in advances in non linear pde and related areas world scientific 1998 pp 67 79 y y li i shafrir blow up analysis for solutions of u veu in dimension two indiana univ math j 43 1994 1255 1270 in brezis li and shafrir s result the constant c depends on the lower bound of the function r x in this paper we remove this restriction and extend the inequality to the case where r x is allowed to have zeros so that it can be applied to obtain a priori estimates for a broader class of equations as we will illustrate in the last section the key to prove this inequality is the analysis of asymptotic decay near infinity of solutions for a corresponding limiting equation in r2 which is interesting and useful in its own right

the conformal plate buckling equation the linear equation 2 u 1 for the infinitesimal buckling under uniform unit load of a thin elastic plate over r 2 has the particularly interesting nonlinear generalization g 2 u 1 where ag e 2u is the laplace beltrami operator for the metric g e 2u g 0 with g 0 the standard euclidean metric on r 2 this conformal elliptic pde of fourth order is equivalent to the nonlinear system of elliptic pdes of second order u x k g x exp 2u x 0 and k g x exp 2u x 0 with x r 2 describing a conformally flat surface with a gauss curvature function k g that is generated self consistently through the metric s conformal factor we study this conformal plate buckling equation under the hypotheses of finite integral curvature f kg exp 2u dx k finite area exp 2u dx a and the mild compactness condition k l 1 b 1 y uniformly with respect to y r 2 we show that asymptotically for x all solutions behave like u x 2 ln x c o 1 and k x 2 ln x c o 1 with 2 4 and 2 4 we also show that for each k 2 4 there exists a k and a radially symmetric solution pair u k satisfying k u k and max k k which is unique modulo translation of the origin and scaling of x coupled with a translation of u

on patlak keller segel system for several populations a gradient flow approach abstract we study the global in time existence of solutions to the parabolic elliptic patlak keller segel system of multi species populations we prove that if the initial mass satisfies an appropriate notion of sub criticality then the system has a solution defined for all time we explore the gradient flow structure in the wasserstein space to study the question of existence moreover we show that the obtained solution satisfies energy dissipation inequality

on a planar liouville type problem in the study of selfgravitating strings abstract motivated by the construction of selfgravitating strings cf yang 2001 1994 22 23 we analyze a liouville type equation on the plane derived in yang 1994 23 we establish sharp existence and uniqueness properties for the corresponding radial solutions we investigate also when the problem allows for non radial solutions

multivortex solutions in the chern simons gauged o 3 sigma model on a doubly periodic domain abstract we establish the existence and uniqueness results for multivortex solutions of an elliptic equation arising from the self dual chern simons gauged o 3 sigma model with a symmetric potential on a flat 2 torus we prove that if the parameter e 0 is small then the elliptic governing equation admits a maximal solution we also study the asymptotic behavior of the maximal solutions as e 0 by using this asymptotic behavior we establish the uniqueness results for solutions which tend to a e as e 0 under suitable conditions on the dirac measure

on the calculation of true and pseudo penalties in multiple choice integer programming abstract the general solution approach leading to the global optimum for multiple choice integer programming uses the branch and bound procedure designed for special ordered sets incorporated in such a solution procedure the partitioning strategy based on a weighted mean method is most often adopted to calculate the pseudo penalties on the partitioned subconstraints for branching and backtracking based on a reformulation and transformation technique we show that the calculation of true penalties of these partitioned subconstraints can be performed efficiently

liouville theorems for kirchhoff equations in rn this paper is devoted to the nonexistence of nontrivial weak solutions for the kirchhoff equation a b rn u 2 dx u x qf u in rn we prove that the equation has no weak solution if a 0 b 0 q 2 and f is a positive convex nondecreasing function if only b 0 and f is a non negative function we establish the nonexistence of weak solutions u satisfying lim infr r1 n2 br x qf u dx 0 this implies that the equation has no weak solution when n 2 and f is a positive function we also show that the equation has no stable weak solution in dimension n 0

singular solutions of parabolic p laplacian with absorption we consider for and the laplacian evolution equation with absorption we are interested in those solutions which we call singular solutions that are non negative non trivial continuous in and satisfy for all we prove the following i when there does not exist any such singular solution ii when there exists for every a unique singular solution that satisfies as also as where is a singular solution that satisfies as furthermore any singular solution is either or for some finite positive

existence and uniqueness of fast decay entire solutions of quasilinear elliptic equations we prove the existence and uniqueness of fast decay solutions and clarify the structure of positive radial solutions of the quasilinear elliptic equation mu f u 0 in rn where m denotes the m laplace operator and f has a supercritical growth for small u 0 and a subcritical growth for large u our proofs use only elementary arguments based on several variational identities and a maximum principle of peletier and serrin

implementations of special ordered sets in mp software abstract special ordered sets sos have been introduced as a practical device for efficiently handling special classes of nonconvex optimization problems they are now implemented in most commercial codes for mathematical programming mp software the paper gives a survey of possible applications as multiple choice restrictions conditional multiple choice restrictions discrete variables discontinuous variables and piecewise linear functions global optimization of separable programming problems alternative right hand sides overlapping special ordered sets and the solution of quadratic programming problems alternative problem formulations are discussed since special ordered sets are not defined uniquely modelling facilities depend on the definition of a special orderedset in a code the paper demonstrates the superiority of sos to the application of binary variables if they are treated judiciously

on the conformal gaussian curvature equation inr2 abstract in this paper we consider the equation u k x e 2u 0 in r 2 0 1 wherek x k x inr2andk x does not decay at x large from the geometric viewpoint it is quite interesting to ask what the best possible range is of total curvature of all solutions of 0 1 in this paper we study this problem for radial solutions we also construct some particularkto demonstrate the rich phenomenon in particular we show by examples that whenk x is negative for x large eq 0 1 possess a branch of solutions which satisfies some monotonicity property